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The performance of robot manipulation van den Doel, Cornelis Pieter 1994

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THE PERFORMANCE OF ROBOT MANIPULATION By Cornelis Pieter van  dell  Doel  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF COMPUTER SCIENCE  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  March 1994  ©  Cornelis Pieter van den Doel, 1994  In presenting this thesis in partial fulfilment of the requirements for degree at the University of British Columbia, I agree that the Library freely available for reference and study. I further agree that permission copying of this thesis for scholarly purposes may be granted by the department  or  by  his  or  her  representatives.  It  is  understood  an advanced shall make it for extensive  head of my that copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  Computer Science  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  April 28 1994  Abstract  A theory of the performance of robot manipulation is presented. The theory deals with general robotic manipulation, which is viewed as the physical interaction between a generalized manipulator and a generalized object. The generalized manipulator can be arbitrarily complex, i.e. it can consist of any number of robotic devices. Similarly, the generalized object represents the set of all the material bodies manipulated by the robot in the environment. The study of the performance of such systems is somewhat analogous to the study of the performance of algorithms and complexity theory in computer science. A digital computer produces certain desired changes in information, it transforms input into output, and complexity theory deals with the efficiency of this process. Similarly, a robotic device transforms the state of the objects it manipulates. The manipulation we are concerned with here is only different in that material objects are affected and the measure for the resources needed to perform certain tasks is not runtime or memory usage. How performance should then be measured instead is the subject of this thesis. The theory regards the manipulation as a mapping from the space of all manipulator configurations to the space of all object configurations. By endowing these manifolds with Riemannian metrics we can quantify motion in the manifolds. Performance measures, which quantify how well the robot is executing its manipulation task, arise naturally as geometrical objects associated with the mapping and the metric structure of the manifolds. Generalizations of known measures (agreeing with those previously defined for the simplest kinds of systems) arise naturally in this formalism and we construct several new types of measures. The new types of measures presented here are the effective inertia on II  work space, the kinematic and dynamic anisotropy measures for redundant manipulators, non-linearity measures, and redundancy measures. An extension of the theory broadens its domain of application to situations in which the relation between the robot and the object it is manipulating can not be given in explicit form, but by a set of constraint equations (possibly non-holonomic) that define it implicitly. The main contribution of this work is the theory of manipulation, which provides a theoretical foundation for robot manipulation. Computational aspects of the theory are described briefly and the theory is applied to some simple planar manipulating devices, for which several of the measures have been constructed and optimized. These applications are given here for illustrative purpose.  111  Table of Contents  Abstract  ii  List of Figures  vi  Acknowledgement  ix  1  Introduction  1  2  The Theory of Manipulation  7  2.1  Model of Robot Manipulation  7  2.2  Construction of Performance Metric and Performance Measures  14  2.2.1  Outline  14  2.2.2  The Performance Metric  2.2.3  Work Space Measures  15  .  .  18  .  2.3  Redundancy Measures  20  2.4  Non-linearity Measures  23  3 Extending the Theory to Constrained systems  28  3.1  Description of Constrained Systems  28  3.2  Construction of the Performance Metric  32  4 Applications to Planar Arms  37  4.1  Kinematic Measures  38  4.2  Inertial Measures  52  iv  4.3  Redundancy Measures  .  72  5  Applications To A Closed Loop Linkage  93  6  Conclusions  99  Bibliography  101  Appendices  104  A Computational Aspects  104  V  List of Figures  1.1  Feasible and not feasible tasks  3  2.2  Relation between robot configurations and manifolds  8  2.3  Construction of induced distance  11  2.4  Relative acceleration of nearby geodesics  25  3.5  Constrained two link planar arm. The end-effector is constrained to remain on the vertical line  3.6  30  Manipulator descriptor space C+ and configuration space C of constrained two-link planar arm  31  4.7  Good posture for applying vertical force  39  4.8  Bad posture for applying vertical force  40  4.9  Good posture for applying horizontal force  .  .  .  41  4.10 Bad posture for applying horizontal force  42  4.11 Yoshikawa measure for 3-joint redundant arm  44  4.12 Optimal postures for Yoshikawa measure for given end-effector position  45  4.13 Optimal values for Yoshikawa measure  46  4.14 Kinematic anisotropy measure for two link arm  47  4.15 Kinematic anisotropy measure A for 3-joint redundant arm  48  4.16 Optimal postures for A  49  4.17 Optimal values for A  50  4.18 Good posture for vertical swatting  54  vi  4.19 Bad posture for vertical swatting  55  4.20 Good posture for horizontal swatting  56  4.21 Bad posture for horizontal swatting  57  4.22 Inertial Yoshikawa measure Y for 3-joint redundant arm  59  4.23 Optimal postures for Y  60  4.24 Optimal values for Y  61  4.25 Inertial anisotropy measure for 2-joint arm  63  4.26 Inertial anisotropy measure for 3-joint redundant arm  64  4.27 Optimal postures for A  65  4.28 Optimal values for A  66  4.29 Curvature scalar R for 3-joint redundant arm  68  2 for 3-joint redundant arm 4.30 R  69  4.31 Optimal postures for R  70  4.32 Optimal values for R  71  4.33 Redundancy measure D for 3-joint redundant arm  .  .  .  73  4.34 Optimal postures for D  74  4.35 Optimal values for D  75  4.36 Distance of end-effector from base d  76  4.37 Area redundancy measure D for 3-joint redundant arm  77  4.38 Optimal postures for D  78  4.39 Optimal values for D  79  4.40 Joint redundancy measure E 1 for 3-joint redundant arm  81  4.41 Optimal postures for E’  82  4.42 Optimal values for E 1  83  4.43 Joint redundancy measure E 2 for 3-joint redundant arm  84  4.44 Optimal postures for E 2  85 vii  4.45 Optimal values for E 2  85  4.46 Joint redundancy measure E 3 for 3-joint redundant arm  86  4.47 Optimal postures for E 3  86  4.48 Optimal values for E 3  87  4.49 Link redundancy measure L for 3-joint redundant arm  89  4.50 Optimal postures for L  90  4.51 Optimal values for L  91  5.52 Closed ioop manipulator  94  5.53 Optimal postures for various orientations of the middle link  98  viii  Acknowledgement  This work was done in collaboration with Dinesh K. Pai. I thank Alan Boulton for useful discussions and help with LATEX.  ix  Chapter 1  Introduction  What is the best way to swat a fly? To hammer a nail? To carry a heavy object? One of the goals of robotics is to understand and automate the manipulation of the physical world by computer. The agents that interact physically with material objects are the robot manipulators. For a given task (a desired change in the state of the objects) one can ask questions such as: “What configuration of the manipulator is optimal?”, “What is the best manipulator for this task?”, and “What class of tasks can be performed by a particular robot?”. The answers to these questions are critical to the design, selection, and programming of robots. A theory of performance is presented, which assigns a numerical value (“performance measure”) to an elementary interaction between the manipulator and the object, within the context of a task. The questions mentioned above can then be translated into opti mization and feasibility problems. This theory is a step towards a complexity theory of robots and robot tasks. Performance measures are already widely used for design and posture optimization for robot arms. The theory presented here has the following novel features: • A unification of measures proposed before in one theoretical framework, based on differential geometry. • A “toolkit” to make your own performance measure suited for the specific manip ulator and task at hand.  1  Chapter 1. Introduction  2  • The construction of promising new performance measures for redundancy and non linearity. • The measures can be used for arbitrarily complex manipulators and objects, unlike existing measures. As a simple illustration of these ideas, consider a robot arm with seven revolute joints and a gripper (end-effector) attached to the end of the arm. The position and orientation of the end-effector has six degrees of freedom, whereas the arm has seven. Such a manipulator is called a redundant manipulator. See for example [Nak9l]. Therefore, if the task consists of grasping an object, there are infinitely many postures to do this. If we have a local performance measure, i.e. a number  t  associated with every posture  (in the context of a specific task), we can resolve this redundancy by finding the posture with optimal 1 u. From a design perspective we can find the optimal design for a set of tasks, by selecting the design with the largest average value of some relevant local performance measure. If this involves a continuous infinity of tasks that the manipulator is expected to perform, we obtain such a design measure by the integration of a local measure over some finite region in space.  Chapter 1. Introduction  Feasible  3  Not feasible  Figure 1.1: Feasible and not feasible tasks  Chapter 1. Introduction  4  Finally, for a given task it is often possible to determine if the manipulator can perform it by examining a performance measure. For example, suppose the task is to pull a nail from a wooden structure with a robot arm. Depending on the location of the nail with respect to the base of the arm, the arm may or may not be strong enough to perform the task. One of the performance measures we shall investigate later can be interpreted as “the minimal sum of the squares of the joint torques needed to apply a (generalized) force with the end-effector”, is that  it  max, t /-  i  where  for short. A necessary condition on the feasibility of such a task max 1 /-  is the sum of the squares of the maximum joint torques  that the motors can provide. So  .t  acts as a complexity measure of the manipulation in  this example. In figure 1.1 we depict two situations. In the leftmost picture there exists a posture that grabs the nail and has have p’>  but in the rightmost picture all postures  ‘  and the task is not feasible.  A complete task will consist of a sequence of elementary tasks. A task planner could utilize local performance measures by examining the feasibility of these subtasks in ad vance. In this thesis we will deal with the construction of local performance measures for various tasks, for generic robotic devices. Several local measures have been proposed in the past. Yoshikawa [Yos85] proposed the scalar /det jjT, where J is the Jacobian matrix of the manipulator. This measures how much the end-effector moves for a given infinitesimal movement of joint angles, averaged over all directions. It is the generalization to redundant manipulators of the determinant of the Jacobian. The related matrix  (JJT)1  when sandwiched between the  work space direction vector and its transpose, measures the analogous manipulability in a particular direction. The inverse, [Chi88].  jjT,  then measures “pushability” in a given direction  Chapter 1. Introduction  5  From the singular value decomposition representation [KL8O, Nak9l] some closely re lated measures can be derived. Salisbury and Craig [SC82] proposed to use the condition number of J, which is the ratio of the largest and smallest singular values of J. Klein and Blaho [KB87} use the minimum singular value as a measure. The distance of the joint angles from their central positions was used by Liegois [Lie77]. Asada [Asa83] computed the effective inertia matrix for a non-redundant manipulator and proposed to minimize the inhomogeneity of the moment of inertia. Angeles and Ma [MA9O, MA93] introduced the dynamical conditioning index, which is a measure for the anisotropy of the moment of inertia of the arm, which was used to design isotropic manipulators. The kinematic isotropy was investigated in [ALC92]. All the measures mentioned above arise naturally in the framework presented here and several new measures are introduced. Care has to be taken to ensure that a constructed measure corresponds to a physical property of the device and is not just a mathemat ical construct. This is achieved here by imposing invariance under general coordinate transformations in configuration space, i.e. we consider measures that do not depend on the actual coordinates (such as the joint angles for revolute joint arms for example) used to describe the configuration of the manipulator. This criterion is sufficient, but not necessary in general, as the coordinates used usually have a direct physical meaning. The approach taken here is to define various metrics on the configuration space, i.e. to define a “distance” between configurations, by a metric tensor. From the forward kinematics of the system we then construct an “induced” metric tensor on the work space, which has its own metric structure. This permits the use of standard differential geometry (see, for example, [DP9Oj) to construct geometrical tensor fields which are interpreted and used as performance measures. The physical interpretation of these measures will depend on the initial choice of the metric on configuration space. For example, the Eudidean metric leads to kinematic  Chapter 1. Introduction  6  measures, while using the inertia matrix as the metric tensor leads to dynamic measures. A mathematical formulation of these ideas are presented in chapter 2. Chapter 3 deals with an extension of the theory to incorporate systems with constraints (such as closed loop kinematic chains, for example). Chapter 4 presents applications of the theory to the planar two and three link manipulators, for which we compute and optimize some of the measures. In chapter 5 we illustrate the use of performance measures for constrained systems for a planar closed loop linkage. Conclusions and directions for future work are presented in chapter 6. In appendix A some computational issues regarding the measure are discussed.  Chapter 2  The Theory of Manipulation  2.1  Model of Robot Manipulation  We model the interaction between a generalized manipulator (any collection of robotic manipulating devices) and a generalized object (the collection of material bodies that are manipulated) as follows. The set of all possible manipulator configurations is described by the configuration space, which we denote by C, and its dimension is n. A point in C represents a configuration of the manipulator, and all points in C represent different configurations. Similarly, all object configurations are described by the work space, which we denote by W. We denote its dimension by m. A task can now be seen as a motion from one point in W (the present state of affairs) to another point. This motion is to be achieved by the manipulator, and we describe the coupling between the two by a mapping C  —*  W  which associates a point in W with every point in C.  For typical robot arms with  a manipulating device attached to the end of the arm, this mapping is the forward kinematics of the system. See figure 2.2.  7  Chapter 2. The Theory of Manipulation  8  C  w  x  q  Figure 2.2: Relation between robot configurations and manifolds  Chapter 2. The Theory of Manipulation  In this section we shall assume that constrained systems, where  i  ic  9  is given in explicit form. The generalization to  is defined implicitly, will be deferred to the next chapter,  mainly because this is technically a little more involved, although there is no conceptual difference. With C and ¾) we associate 1—to—i mappings C  —*  R and ¾)  —*  tm which define lR  charts or coordinate systems on the manifolds. The coordinate system on a manifold is considered here as an arbitrary labeling of the points in the manifold. A point in C is denoted by the coordinates  q’ q= qfl and a point in ¾) by xl  x=  The preimage of a point x in ¾) is the self-motion manifold .A/ C C, which is the set of all configurations that map to this point in ¾’. The tangent space to JV at q is the null-space of the Jacobian of the mapping  ic.  Where it is clear which point x is being  referred to, we will drop the subscript x. The self-motion manifolds are important for constructing redundancy measures, as discussed below. To quantify displacements in the manifolds, we need to define a distance. As the manifolds are non-Euclidean in general, we define a vector norm on the tangent bun dles TC and T¾). This corresponds intuitively to defining the length of infinitely small line segments everywhere in the manifolds. These infinitesimal line segments expand to  Chapter 2. The Theory of Manipulation  10  finite vectors in the tangent space associated with the manifold at the location of the in finitesimal line segment. Distance on C represents the amount of “work” the manipulator has to do for the corresponding motion. It can be defined in many way, which depends on the application at hand. Distance on ¾) represents how much is achieved with the corresponding movement. For example, consider the configuration space of a planar two link arm, with two revolute joints. The configuration space C is isomorph to the two-torus T , or to a section 2 of it if there are limits on the joint angles. A natural coordinate system is provided by the two joint angles, i.e. q =  (q’,  )T. 2 q  An example of a work space is the space of all configurations of a rigid body, SE(3) 3 x SO(3). However, the object that is manipulated may itself be some mechanical 1R device, such as a mechanical arm. The performance can now be described by an “induced” distance on ¾), that describes how much the manipulator has to do in order to achieve a given infinitesimal displacement in ¾). Taking the example of a robot arm with an end-effector attached to the tip, the distance on C measures movements of the arm, the distance on ¾) measures movements of the end-effector and the induced distance on ¾) measures the minimal movement of the arm needed to achieve this motion in ¾). The construction of the induced norm on ¾) proceeds as follows. Let the manipulator be in posture q, with the end-effector at x. See figure 2.3. We define the induced distance Ik’Iinduced from x to x + dx as the length of the shortest path in C (with respect to the chosen metric h) that generates a motion in ¾) from x to x + dx.  Chapter 2. The Theory of Manipulation  11  I  C  w Figure 2.3: Construction of induced distance  Chapter 2. The Theory of Manipulation  A vector norm  l  • Mxli > 0 if x  • lkxIi  =  • Mx +  M  12  has the following properties: 0  cixM < lxii +  lyW.  It is not difficult to prove that the induced norm on TVV, as defined above, satisfies these conditions if the norm on TC does. We now make the additional assumption that the norms on TC and TV1) are quadratic norms, i.e. they can be given by  lxii  v/’xTAx, where A is a symmetric, nondegenerate,  positive definite matrix. The matrix A is the metric tensor of the manifold. A manifold with such a quadratic norm is a Riemannian manifold. (Relaxing the condition of positive definiteness on the metric tensor leads to a pseudo-Riemannian manifold.) A non-trivial result is that if the norm on C is quadratic, then the induced norm is also quadratic. This will be proved by explicit construction below. Tensor indices in C are denoted by lowercase latin letters ranging from 1 to n, in ¾) by greek letters ranging from 1 to m and in A by uppercase latin letters ranging from 1 to n  —  m.  1  The components of tensors in C are w.r.t. the coordinate basis e dual 1-form basis w  =  =  and the  dq. We use the Einstein summation convention for tensor indices  throughout. That is, any repeated tensor index is implicitly summed over its appropriate range. See [DP9O] (or any book on differential geometry) for a detailed description of the tensor formalism. The metric tensor on C is written as h with components h . The (length) 2 2 of an ‘Generically, i.e. if the mapping  is non-singular.  See also [PL92].  Chapter 2. The Theory of Manipulation  13  infinitesimal vector dqe is given by the “line-element” 2 ds  =  . 3 hdqdq  2 ds  =  dqTh dq.  In vector notation this reads  is the covariant (lower indices) form of the metric tensor. The inverse of h has and satisfies  components  =  where 6 is given by 1, ifi=j,  1 The  O otherwise.  are the components of the identity matrix. On ‘V we have a separate “task”  metric tensor  ij  with components  ,  such that  2 ds  =  dx’dx”  is the square of the “task” length of the infinitesimal vector with components dxv. For a typical robot arm, this will be some combination of the spatial distance between points and their difference in orientation in the work space. Note that this metric is non-trivial in general as ¾) is generally non-Euclidean. For example if ¾) is the set of positions and orientations of the end-effector of a three dimensional manipulator W  =  3 ® SO(3), 1R  which is not Euclidean. Besides the task metric we construct an “induced” metric tensor on ¾) (induced by the metric h on C), written as g, with components grn,. We shall call this the “performance” metric on ¾).  Chapter 2. The Theory of Manipulation  2.2  14  Construction of Performance Metric and Performance Measures  2.2.1  Outline  The construction of the measures proceeds as follows. • Define an appropriate configuration space C and a metric tensor h on C. This describes the robots and quantifies their motions. • Define a task space ¾) and a metric tensor  ij  on ¾). This describes the objects to  be manipulated, with an appropriate metric for their motions. • Define a mapping  ic :  C  —*  ¾’. This describes the manipulation of the objects by  the manipulator. • From ,c and h we construct an induced metric on ¾), the performance metric g. This is a tensor valued performance field from which other work space performance measures can be derived. • From g and  ij  we construct performance measures. Note that, in general, a measure  is defined by the tuple {C, ¾’, ,c, h, j}. The measures we consider here are the following. —  A relative directional performance field U. This field depends on a direction specified by the vector u in ¾).  —  A relative average performance field, which we call the generalized Yoshikawa measure Y.  —  A measure of the performance anisotropy of g, the anisotropy measure A.  —  A measure R for the non-linearity of the motion of the manipulator. This is the double contraction of the Riemann tensor of the manifold.  Chapter 2. The Theory of Manipulation  —  15  A class of measures for the reconfigurability of the manipulator (i.e. its ability to move without moving the objects), which we call the redundancy measures.  2.2.2  The Performance Metric  The map Ic : C  —  )/V is defined by XIL =  1c4u(q)  (2.1)  and the Jacobian J of the map is written as jIL(  —  where J(q) are the components of the Jacobian J. The construction of an induced metric on W proceeds as follows. Let the manipulator be in posture q, with the end effector at x. We define the induced distance d(y) from x to y as the length of the shortest path in C (with respect to the chosen metric h) that moves the end-effector from x toy. The metric tensor on )‘V can now be derived by considering the shortest path in C, from q to q + dq, that moves the end-effector from x to x + dx. From eq. (2.1) we obtain dx  =  Jdq.  (2.2)  The minimum norm solution of eq. (2.2) is given by dq=Jdx  (2.3)  where the pseudo-inverse J+ is given by J+  =  h_1JT(Jh_1JT)’.  JT exists. 1 We assume here that the configuration is non-singular, i.e. the inverse of Jh_  Note that this pseudo-inverse depends on the metric h, as this defines the norm. So the  Chapter 2. The Theory of Manipulation  16  induced norm of dx is given by 2 ds  =  dxTJ+ThJ+dx  hdqdq’  and we define g  =  J+ThJ+  as the covariant metric tensor on )‘V. The field g is a tensor valued performance field. g has a simpler contravariant form J,’h J 2 7  =  or, in matrix notation (as gW are the components of g’), =  g’  Jh_ J 1 T.  The minimum norm solution to eq. (2.2) can now be written as dq  =  Jdx’  where J are the components of J and are obtained from J by raising and lowering its indices with the appropriate metric: —  •  This provides an elegant tensor representation of the pseudo-inverse of the Jacobian. Note that g , does not carry any C space indices and is therefore invariant under gen 4 eral coordinate transformations in C. It transforms as a tensor under general coordinate transformations in ¾). Scalar directional performance fields can be obtained from g, by contraction with an appropriate vector u in ¾), i.e.  IWU4UUV. 9  For robot arms, some natural choices for h are the Eudidean metric, which measures 2 as the sum of the squares of the joint angle differences, and the inertia tensor, (distance) i.e. we take h=H,  Chapter 2. The Theory of Manipulation  17  where H is the inertia matrix of the arm. We shall refer to the former as the kinematic metric and to the latter as the inertial metric. With the inertial metric, the geodesics on C are precisely the solutions to the equations of motion of the system in the absence of torques, friction, and external effects such as gravity. Another possible metric is the joint-compliance matrix, which measures the “stiffness” of the arm (e.g., [PL91]). This can be considered as a generalization of the Euclidean metric with different weights for joint distances. We note that the kinematic measures are the same as the measures constructed from a compliance matrix with equal joint compliances. The choice of the metric  2’  on VV is a non-trivial matter in general. For the simplest  kind of object, a dimensionless point in space, the obvious choice for r is the identity matrix (in a Cartesian coordinate frame). But for a rigid body there is no unique metric. The reason is that we want to define the distance between configurations that differ in position and orientation. Position is measured by coordinates, in meters, but orientation is measured by angles, which are dimensionless. To define a combined distance we need some dimensional weight factor that converts between the two. From a mathematical point of view, there exists no hi-invariant metric on SE(3), the group of rigid body motions [Lon85j. A natural metric for a rigid body is constructed as follows. Let dx be the difference in position and let dO be the angle of rotation (around some axis). The metric is defined by 2 ds  =  dxTdx + 2 d0 A ,  with A a length scale. This combines rotation and translation through the length scale  A. Another natural choice would be to take the inertia matrix of the rigid body as the metric tensor. In that case the inertia matrix would provide the necessary dimensional  Chapter 2. The Theory of Manipulation  18  conversion between angles and positions. Indeed, for a sphere with mass M and moment of inertia I, the inertial metric would coincide with the metric mentioned above, with A=I/M. Which is the metric to choose depends on the application at hand. 2.2.3  Work Space Measures  The scalar U with  ,  (2.4)  =  the task metric on ¾), measures task space length per C space length for move  ments in direction u. It is a directional performance field. From g and  i  we derive a scalar measure that we call the generalized Yoshikawa  measure. Consider an infinitesimal parallelepiped dx’... dxm located at the point x in ¾) and suppose the manipulator is in configuration q. The rn-volume of the smallest region in C that will cover the parallelepiped in ¾) is given by Jdet(g)dx1... dx . tm It measures the volume of the minimal region in which the manipulator has to move to cover the given infinitesimal parallelepiped in work space. The task volume of this region is /det(’rj)dx’... dx . tm We define the generalized Yoshikawa measure as the ¾) space volume per C space volume of the parallelepiped,  N  det() det(g)  (2.5)  It is a measure of the performance of the manipulator, averaged over all directions. Note that Y is invariant under general coordinate transformations in C as well as in ¾)  Chapter 2. The Theory of Manipulation  19  space and therefore measures a true physical property of the system, independent of the coordinate systems. 2 If h is the Euclidean metric in “joint-angle” coordinates and the task metric on ¾) is Euclidean, it reduces to the original Yoshikawa measure det(JJT) as defined in [Yos85]. A second scalar derived from the metric tensor g is the anisotropy. It measures how isotropic the manipulator behaves locally. There is a slight complication in defining this measure, as for a general metric 77 there is already an inherent anisotropy present in the work space, which depends on the coordinate system used. We first eliminate this by finding a local coordinate transformation that brings j into Euclidean form, ie. 77 T TT  =  1,  where T is the matrix associated with the coordinate transformation. In these new coordinates the induced metric becomes =  TgTT.  The equation xTx =  1  defines an ellipsoid, with the lengths of the principal axes corresponding to the eigenvalues of  .  The elongation of the ellipsoid (i.e. how much it differs from a sphere) in this special  coordinate system is a physical measure of the anisotropy of the manipulator. It is easy to show that the eigenvalues of  are the same as the eigenvalues of ‘g, in any coordinate  system. The same information is contained in the inverse of this matrix, i.e. g . Since 1 g may have singularities (if the manipulator is in a singular configuration such that it cannot move in some direction in ¾), the metric tensor g will become infinite in that direction) it is better to work with this matrix instead. 2 T he field /det(g) itself is not invariant under general coordinate transformations in 34). It is called a scalar density.  Chapter 2. The Theory of Manipulation  20  We define the invariant measure of the anisotropy A as A=1——,  (2.6)  where i+ and 1 u_ are the largest and smallest eigenvalues of g’. The measure given in eq. (2.6) measures the anisotropy. If isotropy is desired for a task, this measure should be minimized. For positive definite metrics h we have 0 <A < 1 and isotropy is obtained if A  =  0.  We note that the anisotropy measure as defined by eq. (2.6) may depend on the length units used if we consider a work space which incorporates both positioning and orientation information. In this case, the measure A will depend on an arbitrary length scale that corresponds to the relative weight we assign to orientation (described by dimensionless coordinates, e.g. Euler angles) relative to positioning (described by coordinates with the dimension length). The role of such a length scale was investigated in [TAR92]. It emerges here in the form of the factor  i.e. the work space metric. For systems that  illcorporate both positioning and orientation information it is not always possible to find a unique metric  j.  This arises because we somehow have to combine angles and lengths  and we need a dimensional parameter to convert between the two. Other performance measures can be derived as needed from the above by covari ant differentiation and contraction. For example, the local variation of the generalized Yoshikawa measure could be defined by g  2.3  a y 2 ’8q0qJ 1  Redundancy Measures  An important class of measures is related to the ability of the manipulator to move in the self-motion manifold .1V associated with every point x in )‘V.  Chapter 2. The Theory of Manipulation  21  Consider a redundant robot arm that performs some assembly task inside some en closed region by reaching inside, through a small opening. It will be “easier” to reconfig ure the arm (change its posture without moving the end-effector) if the segment of the arm that enters through the opening does not move very much during the reconfigura tion. A possible measure for the “goodness” of the posture for a task like this is the ratio of the C space distance and the displacement of the segment that reaches through the hole (as measured in the plane of the surface with the opening). Redundancy measures quantify the mobility of the manipulator in the implicitly de fined self motion manifold .,V,. For the case of a robot arm, this means that we are effectively considering a linkage with a closed loop, constructed out of the original arm by fixing the end-effector at x. The redundancy measures are obtained by a similar construction as the work space measures. We take the x-dependent tuple =  {C, )‘V,  h, i,}  as the generator of the measures. The tuple T is related to C, ¾) and  Cx so that  i  is the identity map.  t  =  as follows. We take VV =f  (The domain and range of the map depend on x.)  However, we measure “distance” in C and ¾ 4 using different metrics. The metric h on C, is just h restricted to metric  i’  We denote it by n, dropping the subscript x. The task  on W is obtained by restricting a “secondary” metric h to .A4. With these  choices for T it is possible to construct the measures without explicitly constructing .Af as follows.  Chapter 2. The Theory of Manipulation  22  An infinitesimal displacement dq in iV, can be written as dqz  _  d’,b 3 P 2  for arbitrary d’b. The projection operator P is given by  The Jacobian J defines n generate motions in  T14/X  —  m  independent vectors  WA,  (except at singular points). The Jw  =  WA  (A  =  1,.  .  .  ,  ii  —  m)  that  satisfy  0.  An arbitrary vector in TJ\f can be written as v  =  wvA.  Its h-length is hvv’  =  hwwvAvB  and we obtain the components of the metric tensor n on AB  Similarly, the task metric  Thc  As the mapping  =  is obtained by restricting h to 1XAB T  as  i.e.  _..i j JWAWB. —  is just the identity, the Jacobian is just the identity matrix and the  induced performance metric g on W, is just the metric n. With these definitions we can construct all the measures derived previously for the work space W, such as the directional performance U, the generalized Yoshikawa measure Y, and the anisotropy A. By considering all points x and noting that the AI are all disjoint, we finally arrive at  Chapter 2. The Theory of Manipulation  23  measures defined on the entire configuration space C. We call these the “redundancy” measures. The precise interpretation of a measure will depend on the secondary task metric h chosen, which is very task-specific in general. In the examples given in chapter 4, we make some choices for h and interpret the secondary generalized Yoshikawa measures. Other redundancy measures, such as the anisotropy or non-linearity (discussed below) can only be defined if the dimension of .IVX is at least 2. A more general set of redundancy measures (which are just the work space measures of the self-motion manifold) can be obtained by considering non-trivial mappings  ‘,  to a work space W of possibly lower dimension than .A/. We shall not consider those measures here, but they can be constructed easily by using the extension of the theory to constrained systems, which is the subject of chapter 3.  2.4  Non-linearity Measures  The purpose of the measures introduced in this section is to quantify a posture in terms of the “complexity” of the dynamics of the manipulator in a small region in configuration space around this posture.  Robot manipulators obey highly non-linear equations of  motion. One can expect that a control scheme that incorporates some form of numerical integration of the equations of motion will require less resources if the non-linearity of the system can be minimized in some sense. This issue is complicated due to the fact that the equations of motion depend on the coordinate system that is used on configuration space. As explained previously, the equations of motion of a manipulator (in the absence of torques, friction and gravity) are identical to the equations for the geodesics on C with the inertia matrix as the metric tensor h on C.  Chapter 2. The Theory of Manipulation  24  The equations of motion can be written as (2.7) where q are the coordinates on C. The Christoffel symbols —  Fijk  and the  —  1 (oh Ohk + I Oqk 2 \ ôq3  —  —  —  r  are given by  ãhk 8q  are obtained by raising the first index, i.e.  “jk  =  3 I 1 h k ’1 .  The non-linearity of eq. (2.7) can be seen as a result of the presence of the I’jk. However, as this is not a tensor (see for example [DP9O]), the Christoffel symbols themselves cannot be directly used to obtain a physical (i.e. coordinate system independent) measure of the non-linearity. We propose to measure the non-linearity of the manipulator with respect to the metric h on C by the Riemann tensor, which is a rank 4 tensor field on C, with components Rkl. The Riemann tensor has  (2 —  1) independent components. In a coordinate basis it is  given by 9ri.  api. —  Rkl =  ôqk  +  —  “m1’jk  It is well known that a global coordinate transformation can be found that linearizes the equations of geodesic motion if and only if the Riemann tensor vanishes everywhere. This point was stressed for mechanical systems in [Spo92, Bed92j.  Chapter 2. The Theory of Manipulation  25  V  Figure 2.4: Relative acceleration of nearby geodesics  Chapter 2. The Theory of Manipulation  26  The Riemann tensor has the following interpretation. Suppose we have two nearby points in configuration space with (small) separation vector s and these points move in the same direction (i.e. on parallel trajectories), described by a velocity vector v, on geodesic trajectories (see figure 2.4). If the space is non-Euclidean, the trajectories will not remain parallel. They will diverge or converge (at least for generic s and v) and the relative acceleration between the points is given by RujklvJsIvvl.  (2.8)  Therefore, this tensor measures directional non-linearity of the arm. If the tensor van ishes, an error in the position of the arm (in configuration space) remains constant over time, as nearby geodesics remain parallel. The relative acceleration as given by eq. (2.8) is independent of the particular coordinate system used and therefore measures an intrinsic property of the manipulator. There is also an interesting relation between curvature and stability. If the curvature is negative, the geodesics are exponentially unstable (because the effective repelling force is proportional to the separation), as nearby geodesics repel each other, whereas if the curvature is positive we obtain stable orbits. See [Arn89], ap pendix 1, for details. The Riemann tensor is a generalization of the Gaussian curvature of a surface. (If the configuration space is two-dimensional, R has only one independent component, due to its symmetries, which is the Gaussian curvature of the manifold.) The associated curvature scalar, R  =  huiRcjkj  gives an overall measure of the non-linearity.  (2.9)  The curvature R is a scalar field, i.e.  it is invariant under general coordinate transformations in C and therefore measures an intrinsic property of the system. Note that this measure does not depend on the forward kinematics. This measure (or R ) could be used to minimize the error of linear 2 approximations used in controlling the manipulator.  Chapter 2. The Theory of Manipulation  If R  =  27  0 at some point, the configuration space is flat at this point. An interesting  observation is that if there are points in C with R> 0 and with R  <  0, there are (n  —  1)-  dimensional subspaces of C with vanishing curvature. If m <n these subspaces will map onto rn-dimensional regions in }V, so these are regions in ¾) that can be reached with an optimal posture with respect to the curvature measure. We shall call such postures “flat”. A curvature tensor and its contractions can also be defined on ¾), using the per formance metric g. This measures the non-linearity of the motion of the manipulated objects. Note that for a non-redundant manipulator, i.e. m  =  n, we can view the the for  ward kinematics at non-singular points as a change of coordinate system on C. Because R is independent of the coordinate system used, the curvature scalar R will therefore be the same whether constructed from h 3 on C or from g, on W, for non-redundant manipulators.  Chapter 3  Extending the Theory to Constrained systems  3.1  Description of Constrained Systems  Many systems are most easily described in terms of a more general system upon which constraints are placed. For example, a walking machine could be most easily described by the configuration of all its joints and the position and orientation of its main body. If the machine is walking, we have the constraint that its feet should be on the ground. For such systems, the geometric ideas presented in the previous chapter can be used to quantify performance, but the details are a little different. The configurations of the constrained systems we are interested in have a natural description in the form of a set of n coordinates 1 q q= qfl upon which some constraint equations are imposed. The coordinates q can be thought of as representing configurations that may violate the constraints, which usually has a physical interpretation as the opening up of some closed kinematic loop. Following [Lue77], we call the unconstrained coordinates together with the work space coordinates  descriptors. The unconstrained coordinates q by themselves are called the manipulator descriptors. The space of all configurations including the ones that violate the constraints is called the manipulator descriptor space, denoted by C+, of dimension n. The points in 28  Chapter 3. Extending the Theory to Constrained systems  29  C C C are labeled by those coordinates q that satisfy the s constraints. The dimension of C is taken to be 1, and the work space ¾) has dimension m. The descriptor space is x ¾), of dimension n + m. Furthermore, we have n+m  —  S =  1.  For an illustrative example, consider the two link planar manipulator depicted in figure 3.5. The base and elbow joints, with angles 0’ and 02, are actuated and the tip of the arm is constrained to slide along a vertical line, at a distance 1.0 to the right of the base of the arm. The two links are of unit length. In this case the manipulator descriptor space C+ is the 2-torus = {(91,92)  :  <01,02  <ir}  depicted as the shaded area in figure 3.6, and the configuration space C C C+ is the one dimensional region C  = {(01,02)  1 +cos(O’ +02) : —ir/2 <0’ <ir/2,cosO  =  1.0},  which is drawn in figure 3.6. The work space ¾) is a line segment of length Tensor indices in TC+ (the tangent space to Cj are denoted by lowercase Latin letters, tensor indices in TC by uppercase Latin letters and tensor indices in T¾) by lowercase Greek letters.  Chapter 3. Extending the Theory to Constrained systems  30  0  Figure 3.5: Constrained two link planar arm. The end-effector is constrained to remain on the vertical line.  Chapter 3. Extending the Theory to Constrained systems  31  02  0’  Figure 3.6: Manipulator descriptor space C+ and configuration space C of constrained two-link planar arm.  Chapter 3. Extending the Theory to Constrained systems  32  The forward kinematics as well as the constraints on the coordinates q are implicitly given by s constraint equations fA(xq)=0 where A  =  1,.  .  .  (3.10)  s, which can not be solved explicitly in general. If eq. (3.10) is linear  in the work space coordinates x, the forward kinematics of the system can be found explicitly. Note that in the following we don’t use eq. (3.10) directly, but only its differential as given in eq. (3.11) below. By allowing arbitrary factors  f and ffk the entire construction  carries over to non-holonomic systems as well. On ¾) we have a “task” metric tensor  as before.  Besides the task metric we  construct an “induced” metric on ¾) (induced by the metric h on C), written as g, with components gpv. We shall call this the “performance” metric on W. This construction is similar to the one presented in the previous chapter.  Construction of the Performance Metric  3.2  On C we define the distance between q and q + dq by a metric tensor h, i.e. with the line element 2 ds  =  dqdq h . 2  The metric on C C C+ then follows from the above for infinitesimal vectors dq tangent to C. Suppose the manipulator is in a configuration labeled by q (satisfying the constraints), with the end-effector at a point labeled by x in ¾) and that we have defined an appropriate metric on C. The induced distance to a point x + dx in ¾) is defined as the distance of the shortest path in C, starting at q, that takes the end-effector to x + dx, where distance is measured by the metric tensor h on C+.  Chapter 3. Extending the Theory to Constrained systems  33  From the metric on C+ we thus derive an “induced” metric on ¾), as before. We construct it explicitly as follows. Consider an infinitesimal motion dq in C with the induced displacement dx in W From the constraint equations we obtain =  fdx” + ffdq  =  0,  (3.11)  where çA L 1 J —  and çA  JJ  çA —  For general dq lying in TC (the tangent space to C at the point under consideration), there is no vector dx that satisfies eq. (3.11). However, we can find the dx that minimizes the quadratic norm (3.12) (where there is an implicit sum over A) using the pseudo-inverse of f viewed as an s x m matrix. We thus obtain dx  =  2 J’dq  (3.13)  f+IUJA  (3.14)  where the “Jacobian” is given by t Jg  and  are the components of the pseudo-inverse of f+IL_  J  where p is the inverse of  A—P  f,  which is given by  wçA Jv’  i.e. pff  =  6’,  (3.15)  Chapter 3. Extending the Theory to Constrained systems  34  provided that the inverse exists. Eq. (3.15) can be written in matrix form as follows. Let us write F for the s x m matrix with components  f,  and F+ for its pseudo-inverse.  With this notation, eq. (3.15) reads =  FT(FFT)*  We can now obtain an explicit condition on dq by demanding that eq. (3.13) is a solution to eq. (3.11). By substituting eq. (3.13) into eq. (3.11) we obtain the following explicit constraint on dq PAdqi  =  (3.16)  0,  where pAfAJ1LfA  Eq. (3.16) defines s  —  m independent constraints on dq, leaving 1  =  m+n  —  s actual  degrees of freedom. Let us choose a basis for the tangent space TC to C by chosing 1 vectors in TC, A= 1,..  .  ,  WA,  1, such that PAwi =0.  In this basis, the C-space metric tensor is hAB  The Jacobian on C is the restriction of the Jacobian given in eq. (3.14) to TC. It is given by Il’— —  i1  i  WA.  The induced performance work space metric can now be obtained. Note that this metric is a field on C x Vu with its components living in TV9. For a given posture, it quantifies the metric in the work space locally. We define the induced distance d(y)  Chapter 3. Extending the Theory to Constrained systems  35  from x to y as the length of the shortest path in C (with respect to the metric h) that moves the end-effector from x to y. For kinematically redundant manipulators, this distance depends on the posture with which the point x is reached. By considering in finitesimal motions we derive the induced performance metric as follows. An infinitesimal displacement dq in TC can be written as dq  =  WAdr,  where the drA are the components of dq in the basis for TC.  From eq. (3.14) and  eq. (3.17) we obtain dx  =  JdrA.  (3.18)  The induced length of the vector dx is now defined as the length of the shortest (under the metric h) vector dr that satisfies eq. (3.18). Eq. (3.18) reads in matrix form dx=Jdr and the weighted pseudo-inverse J+, J+  =  h_1JT(Jh_1JT)’  gives the minimum norm solution to eq. (3.19) in the form dr  =  Jdx.  The norm of dr now gives us the line element 2 ds  =  hABdTAdTB  =  dxTJ+ h T J+dx  from which we read off the performance metric tensor in covariant form as g  =  J+ThJ+.  (3.19)  Chapter 3. Extending the Theory to Constrained systems  36  The inverse of g, corresponding to the contravariant form of g (i.e. g) has the simpler form =  1 g  JT. 1 Jh_  In component notation this reads g  —  —  pLAB z, ‘A”’ ‘B  It defines distance in ¾) as derived from minimal motion in C. All the performance measures can now be constructed as for the unconstrained sys tems.  Chapter 4  Applications to Planar Arms  In this section we shall construct and interpret some of the measures for the two and three link planar manipulators. We shall consider two metrics on C, the Euclidean metric 2 as the sum of the squares of the joint angle differences, and which measures (distance) the inertia tensor, i.e. we take h=H  (4.20)  where H is the inertia matrix of the arm. We shall refer to the Euclidean metric as the kinematic metric and to the metric defined by eq. (4.20) as the inertial metric. The two link arm consists of two links of length 1 and 12. We take all mass to be concentrated in the middle joint and in the tip of the second link, with masses m 1 and 2 respectively. The configuration of the arm is described by the two joint angles q m 1 and , with —r <q 2 q  7t.  The three link arm consists of three links of length I  (  i  =  1, 2, 3) which we take to  be thin rods of uniform mass density m. The configuration of the arm is described by the three joint angles q, with —r < q t < ir. Note that the non-directional measures do not depend on the first joint angle q’ because of rotation invariance, hence it is not shown in plots of the values of the measures. Unless given explicitly in a formula, we take all lengths to be equal to one and all masses to be equal to 0.5 in our calculations.  37  38  Chapter 4. Applications to Planar Arms  4.1  Kinematic Measures  In this case the induced metric on ¾), g, leads to Yoshikawa’s [Yos85] manipulating force ellipses, defined as the set of all forces f/A satisfying gff  1  and the associated manipulability ellipses, defined as the set of all end-effector velocities v that satisfy 1. For a desired end-effector force f L, the measure 1 ç g pvcJ uJv 1  equals the Euclidean norm of the generalized joint torque needed to apply a prescribed force f with the end-effector. Denoting the joint torque by equilibrium is T  —  fJ  and it follows that =  =  3 ffJJ’h  =  Ti,  the condition of static  Chapter 4. Applications to Planar Arms  Figure 4.7: Good posture for applying vertical force  39  Chapter 4. Applications to Planar Arms  Figure 4.8: Bad posture for applying vertical force  40  Chapter 4. Applications to Planar Arms  Figure 4.9: Good posture for applying horizontal force  41  Chapter 4. Applications to Planar Arms  11111111111  Figure 4.10: Bad posture for applying horizontal force  42  Chapter 4. Applications to Planar Arms  43  In figures 4.7, 4.8, 4.9, and 4.10 we show the best and worst postures of the three link arm for applying a vertical and horizontal force with the end-effector at a distance of 2.0 from the base of the arm. The generalized Yoshikawa measure, as defined in eq. (2.5), reduces to the Yoshikawa measure [Yos85] if the factor det() is constant and equal to one as is the case for the planar positioning manipulators we are considering. For easy reference we reproduce here some of the results from [Yos85] for the planar manipulators. For the two link arm we have Y  =  ) 2 l1l2 sin(q  and for the three link arm we have =  1112  —  +  l2 12 13 l2 12  l2  13 + 12 132 + 11 12 13 cos(q ) 2  ) 3 cos(q  —  12  132  13 cos(2 2 q +q ) 3  —  cos(2 q ) 3  —  l2  cos(2 q)  12  l2 132 cos(2  q3)) (q +  + —  cos(q + 2 q ). 3 11 12 132 2  In figure 4.11 we have plotted Y as a function of the joint angles q 2 and q . Figure 4.12 3 shows the optimal postures for reaching a point at a distance of d  =  {0.5, 1.0,1.5,2.0, 2.5}  from the base of the arm and figure 4.13 shows the values of Y at these optimal postures as a function of the reach d.  I’  Ct  F-i.  F-’-  H 01  01 H  0  tO  UT  tO  Chapter 4. Applications to Planar Arms  45  11111111111  Figure 4.12: Optimal postures for Yoshikawa measure for given end-effector position  Chapter 4. Applications to Planar Arms  7  46  I  I  I  6-  5-  4>1  3-  2-  1  0  0  I  I  I  I  I  0.5  1  1.5  2  2.5  a  Figure 4.13: Optimal values for Yoshikawa measure  3  Chapter 4. Applications to Planar Arms  1  47  I  0.95  0.9  0.85  0.8  0.75  0.7  0.65  -  -  -  -  0.6 1 -p  1  0 q2  Figure 4.14: Kinematic anisotropy measure for two link arm  1 p  Chapter 4. Applications to Planar Arms  48  A  0.9 0.8 0.7 0.6 0.5 0.4  0.3  0.2 0.1  1”‘  , -I, II  I III  ‘SI  ‘\4.\ \‘\\‘ \\Fft  ‘‘ I I III  s  ‘II  1 p  0  1 -p  q3  0 q2  -  1 p  -p1  Figure 4.15: Kinematic anisotropy measure A for 3-joint redundant arm  Chapter 4. Applications to Planar Arms  liii  Figure 4.16: Optimal postures for A  49  Chapter 4. Applications to Planar Arms  1  I  50  I  0.9 0.8 0.7 0.6 4:  0.5 0.4 0.3 0.2 0.1  0  0  0.5  I  I  1  1.5 d  2  Figure 4.17: Optimal values for A  2.5  3  Chapter 4. Applications to Planar Arms  51  The anisotropy measure A, as defined in eq. (2.6), measures the deviation from kine matic isotropy. It is zero if the manipulability and force ellipses become circles.  In  figure 4.14 we plot A as a function of the second joint angle q 2 for the two link arm. Figures 4.15, 4.16, and 4.17 show the graph of A for the three link arm, its optimal postures and the values of A for these postures. There is a completely isotropic posture (A  =  0) at a reach of d  =  1.  Note that if the arm moves inward from a stretched position, there is a discontinuity in the optimal arm posture between d  =  1.0 and d  0.5. A more detailed investigation  (a more dense set of postures) shows that the critical distance d is located at d  0.5.  The discontinuity arises as follows. For a given position x of the end-effector, the possible postures form the self-motion manifold JV. The optimal posture is found as the global minimum (or maximum) of the measure on .A1. If for a certain x there are two global minima  ,  an infinitesimal displacement dx of x can resolve the degeneracy. It will depend  on the direction of dx in general which of the two minima will become the new global minimum. It follows that a continuous motion that optimizes the measure through x is not possible is general. An interesting observation is that since .N is not necessarily connected, a motion may be required from one connected component to the other, which is not possible. In that case there is no self-motion possible at all that optimizes the measure. For the three link arm, the connected components of .Af,, which appear for _q As the measures we consider all observe d < 1, are related by the symmetry qZ this symmetry, there will always be two symmetry-related global minima. By picking the appropriate one, the arm is thus never forced to move from one connected component to the other, so that self-motion optimizing the measure is guaranteed to be possible. This is true for the ¾) space measures, but not for the redundancy measures in general, for ‘We refer here to “accidentally” degenerate minima that are not related to each other by a symmetry of the system.  Chapter 4. Applications to Planar Arms  52  the task space and therefore the task metric h need not have the same symmetries as the manipulator (though it does in all our examples). The Riemann tensor and therefore the curvature measure R, as defined in eq. (2.9), vanishes for the kinematic metric on C, as the  4.2  are constant.  Inertial Measures  The generalized inertia matrix H of the system is given by —  H  ôEk(q, i)  —  where Ek is the kinetic energy of the arm. We take the C space metric as =  23 H .  The line-element 2 ds  =  2q h d 2 dq 3 3  measures the displaced mass over the distance dq in C. If the arm moves by dq in a time dt with constant velocity, the kinetic energy of this motion is given by 2 ds  Ek=.  The geodesics in C are now precisely the motions of the arm in the absence of torques, friction and gravity. The induced metric g can be interpreted as an effective inertia matrix on W. That is, if the end-effector moves with velocity x in such a way as to minimize the total kinetic energy of the arm, this kinetic energy is given by Ek  =  Chapter 4. Applications to Planar Arms  53  The tensor g can be measured, in principle, by applying forces to the end-effector, with the arm at rest, and recording the resulting accelerations of the end-effector. For if a force  f, is applied to the end-effector, with the arm at rest, the ‘V space acceleration â  satisfies =  -?  J? +  =  Ji  (4.21)  The equations of motion give =  =  (4.22)  Combining eq. (4.21) and eq. (4.22) gives =  JJ7f 3 h  or c  Jii  Contracting gpv with a velocity vector  —  wX 1 9  VIL  to move the end-effector with this velocity.  gives the required kinetic energy of the arm  Chapter 4. Applications to Planar Arms  Figure 4.18: Good posture for vertical swatting  54  Chapter 4. Applications to Planar Arms  rrni 11111111  II  Figure 4.19: Bad posture for vertical swatting  55  Chapter 4. Applications to Planar Arms  rrn 11111111111  Figure 4.20: Good posture for horizontal swatting  56  Chapter 4. Applications to Planar Arms  11111111111  Figure 4.21: Bad posture for horizontal swatting  57  Chapter 4. Applications to Planar Arms  58  In figures 4.18, 4.19, 4.20, and 4.21 we show the best and worst postures for achieving a high vertical and horizontal velocity of the end-effector (“fly-swatting”) at a distance d  =  2.0 from the base of the arm. The generalized Yoshikawa measure as given in eq. (2.5) is now the inverse of the  square root of the determinant of the effective inertia matrix; it measures the “lightness” of the arm. For the two link arm it is given by ) 2 sin(q s m ( q2)) in +2  I-’.  CD  t’3  a  aq  I-i-  It,  H-  It,  It, I-J  I-I-  It,  P Iii F—s  tfl  CD  cn  Chapter 4. Applications to Planar Arms  rrrrr rrrn  11111111111  Figure 4.23: Optimal postures for Y  60  Chapter 4. Applications to Planar Arms  61  2.4  I  2.35  -  2.3  2.25 >1  2.2  -  2.15  2.1  -  2.05 0  0.5  1  1.5 d  2  Figure 4.24: Optimal values for Y  2.5  3  Chapter 4. Applications to Planar Arms  62  In figure 4.22 we plot this measure for the three link arm. Due to the relatively flat ridges at q 3  =  bir/2, there is a large range of postures with good values of this measure  as can be seen in figures 4.23 and 4.24. There is a critical distance d is a discontinuity in posture as a function of the reach.  1.5 where there  Chapter 4. Applications to Planar Arms  1  63  I  0.95  0.9  0.85  0.8  -  -  -  0.75 -pi  I  0  q2  Figure 4.25: Inertial anisotropy measure for 2-joint arm  pi  B  CD  0  C-”  SAZ  CD  Cl)  BCD  0  0  Cl)  I-.  CD  CD  I-.  H-  It, H-  •  co Ui  0 •  •  0  o ‘D  •  0 Ui  0  1;  Chapter 4. Applications to Planar Arms  Figure 4.27: Optimal postures for A  65  Chapter 4. Applications to Planar Arms  66  0.795  I  0.79 0.785 0.78 0.775 0.77 0.765 0.76 0.755 0.75 0.745 0.74  0  I  I  I  I  0.5  1  1.5 d  2  Figure 4.28: Optimal values for A  2.5  3  Chapter 4. Applications to Planar Arms  67  In figure 4.25 we show the inertial anisotropy, as defined in eq. (2.6) for the two link arm. Figures 4.26, 4.27, and 4.28 show this measure for the three link arm. It also has two valleys at q 3  =  +ir/2, indicating the existence of a range of “good” postures for inertial  isotropy. As the shape of the valleys is different than that of the the ridges for Y, the optimal postures are quite different though. Note that there is no isotropic point in ¾). The critical distance is at d  1.7.  The curvature scalar R as given in eq. (2.9) is now a measure of the geodesic deviation of orbits in C space and is a measure of the non-linearity of the motion of the arm. For the two link arm it is given by R=  cos(q 1 2m ) 2 (mi + 2 2 1 s m ( q)) in  2  Chapter 4. Applications to Planar Arms  68  R  30 20 10 0 —10 —20  0  q3  0 q2  —  Figure 4.29: Curvature scalar R for 3-joint redundant arm  Chapter 4. Applications to Planar Arms  69  R2  1000  500  1 p  -pi  0 0 q2 1  Figure 4.30: R 2 for 3-joint redundant arm  q3  Chapter 4. Applications to Planar Arms  1111111111  Figure 4.31: Optimal postures for R  70  Chapter 4. Applications to Planar Arms  2  71  I  I  I  0.5  1  1.5 d  1.5  1•  0.5  0  —0.5  I  0  2  Figure 4.32: Optimal values for R  2.5  3  Chapter 4. Applications to Planar Arms  72  The curvature scalar is plotted for the three link arm in figure 4.29. In figure 4.30 we show R 2 which is the actual measure to minimize for maximal linearity of motion. As R is positive near the origin of the (q 2 is zero on a ,q 2 ) plane and negative at the edges, R 3 two-dimensional surface in C, where R changes sign. So there is a two-dimensional region in VV that can be reached with “linear” postures. R has local extrema at the singular points (0,0) (R  =  38.1), (0,ir) (R  =  —21.5) and (7r,7r) (R  =  —16.3), but not at (ir,0).  It is interesting to note that this gives a classification of the singular points according to their “non-linearity”, as the values of the extrema are distinct. The optimal postures and values of R are shown in figures 4.31 and 4.32. The critical distance is d 4.3  1.5.  Redundancy Measures  Let us choose the inertial metric as the metric h on C, and the kinematic metric as the “secondary” task metric h. The redundant generalized Yoshikawa measure measures the Euclidean distance per mass moved in C for self-motions. That is, if Y is small, a “large amount of matter” has to be moved to change the posture of the arm. Y should thus be maximized for optimal mobility in this sense. As the inverse D  =  1/Y behaves better  than Y itself we have shown data for D instead of Y itself. The data for this measure for the three link arm is given in figures 4.33, 4.34, and 4.35. The optimal values of D as a function of d (i.e. the distance of the end-effector from the base of the arm) is relatively constant for d> 1.0 but increases fast below that value. In figure 4.36 we have plotted d as a function of (q, q ), from which we see that 3 at small d the arm will come near the large peaks in D, which explains this behavior. The critical distance is at d  1.1.  Chapter 4. Applications to Planar Arms  73  D  1.5  1  0.5  -pi  0  -  q3  q2 1  Figure 4.33: Redundancy measure D for 3-joint redundant arm  Chapter 4. Applications to Planar Arms  11111111111  Figure 4.34: Optimal postures for D  74  Chapter 4. Applications to Planar Arms  1.2  75  I  I  I  I  0.5  1  1.5  I  1  0.8  0.6  -  0.4  -  0.2  -  0 0  a  2  Figure 4.35: Optimal values for D  2.5  3  CD  UI  0 )-  Ui  )  M 01  I  Chapter 4. Applications to Planar Arms  77  D-area 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 1 p  0 1 -p  0  q3  0 q2  .  P1  Figure 4.37: Area redundancy measure D for 3-joint redundant arm  Chapter 4. Applications to Planar Arms  III  111111  Figure 4.38: Optimal postures for D  78  Chapter 4. Applications to Planar Arms  79  1.2  1.1  1  0.9  0.8  0.7  0.6  0.5  0.4  0.3 0  0.5  1  1.5 d  2  Figure 4.39: Optimal values for D  2.5  3  Chapter 4. Applications to Planar Arms  80  We would like to interpret this measure as a measure of how much the arm “swings” when reconfiguring. A better measure for this would be the area that the arm sweeps out per Euclidean distance in C. Unfortunately, the “swept-area” (or volume for threedimensional manipulators) metric cannot be described by a quadratic form, so this type of measure falls outside our framework. For comparison we have computed this “swept area” version of the redundancy measure D. It is given in figures 4.37, 4.38, and 4.39. We observe that this measure behaves qualitatively similar, which supports the interpretation of D given above. A different class of redundant generalized Yoshikawa measures can be obtained by choosing the “secondary” task metric on C as 1, ifi=j=e,  (  0, otherwise.  This metric just measures the movement of one particular joint, e and the corresponding redundancy measure D  =  1/Y will measure the amount of mass moved per movement of  joint e. This could be useful if we want to find an arm position that allows reconfiguration with minimal movement of a particular joint. We call these measures (there are n of them for an  ii  link arm) the joint redundancy measures = y(e)  E(e) measures how much joint e has to move to move a specified amount of mass. It should be minimized if we want to reconfigure the arm with as little movement as possible of this particular joint.  Chapter 4. Applications to Planar Arms  81  El I’, ,<I,;c’  A  )(  1 ‘  o<’,: \-\-*—ç 1<  A  ,  -< ,  0.5  I  -k  I  I  A  V  ,  )W  1 p  0  1 -p  q3  0 q2 1 p  Figure 4.40: Joint redundancy measure E’ for 3-joint redundant arm  Chapter 4. Applications to Planar Arms  ri-fl-I II  111111  Figure 4.41: Optimal postures for E’  82  Chapter 4. Applications to Planar Arms  0.7  83  I  I  0.6  0.5  0.4  0.3  0.2  0.1  I  0  0  0.5  1  1.5 d  I  I  2  2.5  Figure 4.42: Optimal values for E 1  3  I-  CD  0 I-.  C-.  ‘-I  N)  t:xj  CD  CID  CD  CD  0  C3  CD  I—i  I-j.  I-j.  C I-  L  IN)  L.)  00  Chapter 4. Applications to Planar Arms  85  Figure 4.44: Optimal postures for E 2  0.012  I  I  I  I  I  I  0.5  1  1.5  2  2.5  I  0.01  0.008  0.006  0.004  0.002  0  0  a  Figure 4.45: Optimal values for E 2  3  Chapter 4. Applications to Planar Arms  86  2.5 2 1.5 1 0.5  pi  1 -p  1  -p1  Figure 4.46: Joint redundancy measure E 3 for 3-joint redundant arm  Figure 4.47: Optimal postures for E 3  Chapter 4. Applications to Planar Arms  0.3  87  I  0.25  0.2  0.15  0.1  0.05  0  I  0  0.5  1  1.5 d  2  2.5  Figure 4.48: Optimal values for E 3  3  Chapter 4. Applications to Planar Arms  88  Plots, optimal postures and optimal values of the joint redundancy measures are given in figures 4.40  -  4.48. The critical distances are d  1.1  ,  d  0.9 and d  1.0 for  3 respectively. 2 and E E The last example of a secondary generalized Yoshikawa measures can be obtained by choosing the “secondary” metric on C according to 2 ds  =  Euclidean displacement of a test point on a link.  Chapter 4. Applications to Planar Arms  89  L  0.7[0.6 0.5  ‘  i 0.2 0.1  II  1 p  0  1 -p  q3  0 q2 P1  Figure 4.49: Link redundancy measure L for 3-joint redundant arm  Chapter 4. Applications to Planar Arms  Figure 4.50: Optimal postures for L  90  Chapter 4. Applications to Planar Arms  91  0.45  0.4  0.35  0.3  ‘-  0.25  0.2  0.15  0.1  0.05  0  I  I  I  0.5  1  1.5 d  2  Figure 4.51: Optimal values for L  2.5  3  Chapter 4. Applications to Planar Arms  92  For concreteness, we take the test point to be the mid-point of link two. This metric measures the movement of the center of the middle link and the corresponding redun dancy measure D  =  1/Y will measure the amount of mass moved per movement of this  link. This could be useful if we want to find an arm position that allows reconfiguration with minimal movement of this link. We call this measure the link redundancy measure L=Y. L measures how much the middle link has to move in order to move a specified amount of mass. It should be minimized if we want to reconfigure the arm with as little movement as possible of this particular link. Plots, optimal postures and optimal values of the link redundancy measure are given in figures 4.49, 4.50, and 4.5L The critical distance is d  1.5.  Chapter 5  Applications To A Closed Loop Linkage  The closed loop planar kinematic chain we shall investigate here is depicted in figure 5.52. It consists of five links, four actuated joints (6k, 8, 6, O) and two passive joints (O, Os). The end-effector is attached to the center of the middle link. We denote its position and orientation by the work space coordinates (x, y, a). The distance between joints L’ and 1 is d and the length of the middle link is b. The other four links are taken to have unit R length. The descriptor space of the device is labeled by the four joint angles (8k, O, 8, O) which must satisfy one constraint on C+ (or equivalently four constraints on  x ¾)).  We shall take the Cf-space metric to be given by the line element  2 d.s  —  2+2 (d6) (d8j + 2 (dO) + (d8) , 2  i.e. the kinematic metric. This metric defines the distance between configurations in terms of the actuated joint movements. The chain can be interpreted as an object with configuration (x, y, a) manipulated by two planar fingers with joint angles  (, ) and  (6k, 8) with contacts at P1. and PR.  We note that the device has a dual interpretation as a standing posture of a walking machine. The joints 8 and 6 will be passive in this case, representing the feet of the machine and the other four joints will be actuated. In this case an appropriate choice of metric will involve those actuated joints.  93  Chapter 5. Applications To A Closed Loop Linkage  :::::ZEEENL.....l............. (x,y)  /  GE  0> Figure 5.52: Closed loop manipulator.  94  95  Chapter 5. Applications To A Closed Loop Linkage  The coordinates of the passive joints  and  PL  PR  are given by  , 2 p =—d/2—c—c, y  1 12 —SLmSL, —  PL px  =d/2+ck+c,  y  1  —  PR  12  —SRmSR,  where we have written 1  CL 1  —  —cosvL, —  CR  —COSVR,  2  =cos(8.+8,),  c =cos(O+O),  and similarly for the sine functions. The constraints and forward kinematics of the system are given by the four equations fA where A  =  1,..  .  ,  4  ,  =  and ci  .i  2 f  ( x  x\2  “  y  y’2  —PRPLI mkPRPL) —  j2 ,  =(p—p)tana—(p—pj,  3 f 4 f  )/2. 2 =y—(s+s+s+s’,  For this simple system, the constraint on the four joint angles is given by the forward kinematics is defined by the other three equations.  f’  =  0, whereas  Chapter 5. Applications To A Closed Loop Linkage  The three vectors  WA,  A  =  1,..  .  ,  1 = W  3  ,  are chosen as  Of’ Of’ (1,0,0,—-/--), ’L 1  2 = W  96  UUR  Of’ Of’ (0,1,0,—-/-), L  R  and 3 = W  (1,0,0,  Of’ Of’ R  R  The performance metric, as given in the previous section can now be computed with straightforward algebra. To compute the generalized Yoshikawa measure as given in eq. (2.5), we need to define a task metric on VV. Note that since we are considering both the position and the orientation of the end-effector, the definition of a nondegenerate metric on )‘V requires the introduction of a parameter A which is a characteristic length scale. This length scale needs to be introduced as the Euclidean group does not have a natural positive definite metric, see for example [Lon851. A similar length scale was arrived at via a different route in [TAR92]. We take the task metric to be defined by the line element 2 ds  =  2. dx +A +dy d 2 o  The value of A determines the relative weight we attach to orientation versus position in defining a distance between end-effector positions. From eq. (2.5) we see that the factor det(j)  =  A  just multiplies the inverse of the square root of the determinant of the performance metric g. The generalized Yoshikawa measure is thus only defined up to a constant multiplicative factor.  Chapter 5. Applications To A Closed Loop Linkage  97  We have computed Y explicitly as a function of the four actuated joint angles using Mathematica [Wo191j. Since the resulting expression is too complicated to be given here, we have reduced it for illustration to configurations with x  =  0 and c  =  0, corresponding  to symmetric postures, with the center of the middle link above the midpoint between the two base joints. There is one remaining degree of freedom, say 6 , and if we also take b  =  d, we have nl  ‘-‘L  _1  —  —VR—V,  6 =O=ir—26,  and for Y we obtain —  ) 2 (26) sin d 5(1 cos(20)) —  The measure has a maximum value for 6  =  35.3°, which represents the optimal posture  under the given constraints. In figure 5.53 we show optimal postures for configurations where the center of the middle link is above the midpoint between the base joints (i.e. at x ci is restricted to have the fixed values ci  =  0., 0.5, 1.0, 1.5.  =  0) and the angle  Chapter 5. Applications To A Closed Loop Linkage  a  Figure 5.53: Optimal postures for various orientations of the middle link.  98  Chapter 6  Conclusions  The principal merit of the unified formalism for the construction of performance measures for robot manipulators presented here, is that it allows the systematic construction of invariant measures appropriate for a given application. Well known measures such as the Yoshikawa measure arise naturally in this frame work.  In addition, new performance measures have been constructed using this for  malism, to measure non-linearity and redundancy (or self-motion). The formalism was presented for systems in which the relation between the manipulator and the object can be given explicitly. The extension to constrained systems was presented in a separate chapter. Examples of the measures were computed for some manipulators, the planar two and three link arms, and a planar five link closed linkage chain and we computed some optimal postures for these devices. An interesting property of all measures of the three link arm except the Yoshikawa measure is that there is no smooth motion possible from a stretched position (with the end-effector far from the base) to a position with the end-effector close to the base, that optimizes the measure at each point on the trajectory. And for redundancy measures that break the symmetry relating the connected components of the self-motion manifold, it is possible to have end-effector trajectories for which no such motion is possible at all. The curvature scalar measure R that we introduced seems particularly interesting for redundant manipulators, as it identifies an rn-dimensional region in work space where 99  Chapter 6. Conclusions  100  the measure vanishes and the arm behaves locally “linear”. This measure also identifies singular points and classifies them according to their curvature. We note that a singular configuration with a large positive curvature is stable in the sense that geodesics in C attract each other (as the manifold has positive curvature). Unstable configurations arise for negative R, where nearby orbits repel each other and diverge. The measures that were constructed here are by no means the only ones. For example, one could also choose the joint-compliance matrix [PL91] as the metric on C, with a set of associated measures. And the secondary metric on C for the redundancy measure could be any task specific function. While this work was being completed, an article appeared [PB94] in which a similar geometric approach to performance manipulation was taken. The measures examined there were different however, as the authors concentrated on global performance criteria rather than on local ones. Future work will focus on the implementation of the theoretical framework into an automated computation of performance measures from a specification of the system at hand. An important problem that needs to be solved is an examination of the compu tational complexities of the performance measures. In particular the curvature scalar is difficult to compute, as it seems to require 0(n ) operations (with n the dimension of 4 the space), at least in a naive implementation where all the 0(n ) components of the 4 Riemaun tensor are computed.  Bibliography  [ALC92]  J. Angeles and C. Lopez-Cajun. “Kinematic isotropy and the conditioning index of serial manipulators.”. The mt. J. of Robotics Research, 11(6):560— 571, 1992.  [Arn89]  V. I. Arnold. Verlag, 1989.  [Asa83]  H. Asada. A geometric representation of manipulator dynamics and its application to arm design. “J. Dyn. Sys., Meas., Contr. Trans. ASME”, 105(3):131—135, 1983.  [Bed92j  Nazareth S. Bedrossian. Linearizing coordinate transformations and ne mannian curvature. In Proceedings of the 31st Conference on Decision and Control, Tucson, Arizona, pages 80—85, 1992.  Mathematical Methods of Classical Mechanics. Springer-  [BGMBN9O} Angelika Bunse-Gerstner, Volker Mehrmann, Ralph Byers, and Nancy K. Nichols. Numerical computation of an analytic singular value decompo sition of a matrix valued function. In Presented at the Householder XI meeting in Tylösand, Sweden, 1990.  mt.  [Chi88]  S. L. Chiu. Task compatibility of manipulator postures. Robotics Research, 7(5): 13—21, 1988.  [DP9O]  C. T. J. Dodson and T. Poston. Tensor Geometry, The Geometric View point and its Uses. Springer-Verlag, second edition, 1990.  [KB87]  C. A. Klein and B. E. Blaho. Dexterity measures for the design and control of kinematically redundant manipulators. The mt. J. of Robotics Research, 6(2):72—83, 1987.  [KL8O]  V. C. Kiema and A. J. Laub. The singular value decomposition: its compu tation and some applications. “IEEE Trans. Auto. Control”, Ac-25(2):164— 176, 1980.  [Lie77)  A. Liegeois. Automatic supervisory control of the configuration and be haviour of multibody mechanisms. IEEE Trans. Sys., Man., Cyber., SMC 7(12):868—871, 1977.  101  The  J. of  Bibliography  102  [Lon85]  Josip Loncaric. Geometrical Analysis of Compliant Mechanisms in Robotics. PhD thesis, Harvard University, 1985.  [Lue77]  David G. Luenberger. Dynamic equations in descriptor form. IEEE Trans actions on Robotics and Automation, 22(3):312 321, June 1977. —  [MA9O]  On Ma and Jorge Angeles. The concept of dynamic isotropy and its appli cations to inverse kinematics and trajectory planning. In Proceedings of the 1990 IEEE International Conference on Robotics and Automation, pages 481—486, 1990.  [MA93}  Ou Ma and Jorge Angeles. Optimum design of manipulators under dy namic isotropy conditions. In Proceedings of the 1993 IEEE International Conference on Robotics and Automation, pages 470—475, 1993.  [MK89]  Anthony A. Maciejewski and Charles A. Klein. “The Singular Value De composition: Computation and Applications to Robotics”. The Int. J. of Robotics Research, 8(6):63—79, 1989.  [Nak9lj  Y. Nakamura. Advance Robotics: Redundancy and Optimization. AddisonWesley, 1991.  [PB94j  Frank C. Park and Roger W. Brockett. “Kinematic Dexterity of Robotic Mechanisms”. The mt. J. of Robotics Research, 13(1):1—15, 1994.  [PL91j  Dinesh K. Pal and M. C. Leu. Uncertainty and compliance of robot ma nipulators with applications to task feasibility. The mt. J. of Robotics Research, 10(3):200—213, 1991.  [PL92]  Dinesh K. Pai and Ming C. Leu. Genericity and singularities of robot manipulators. IEEE Transactions on Robotics and Automation, 8(5):545 559, October 1992.  —  [SC82]  J. K. Salisbury and J. J. Craig. Articulated hands: Force control and kinematic issues. The mt. J. of Robotics Research, 1(1):4—17, 1982.  [Spo92]  Mark W. Spong. Remarks on robot dynamics: Canonical transformations and riemannian geometry. In Proceedings of the 1992 IEEE International Conference on Robotics and Automation, pages 554—559, 1992.  [TAR92]  Murat Tandirci, Jorge Angeles, and Farzam Ranjbaran. The charac teristic point and the characteristic length of robotic manipulators. In Robotics, spatial mechanisms, and mechanical systems: presented at the 1992 ASME design technics conferences 22nd Biennial Mechanisms Con ference, Scottsdale, Arizona, pages 203—208, 1992. —  Bibliography  103  [Wo191]  S. Wolfram. Mathematica, a System for Doing Mathematics by Computer. Addison-Wesley, second edition, 1991.  [Yos85j  T. Yoshikawa. Manipulability of robotic mechanisms. Robotics Research, 4(2):3—9, 1985.  The  mt.  J. of  Appendix A  Computational Aspects  In chapters 4 and 5, applications of the formalism to some simple planar manipulators were presented. For such low dimensional systems, one need not worry too much about numerical stability and efficiency of the computations. Most of the results presented there were calculated by symbolic calculations in Mathematica ([Wo19 1]), with the exception of the non-linearity measure, which was computed by numerical differentiation from the configuration space metric. In the following, we will briefly describe a different method by which the measures for the unconstrained system could be computed, based on the singular value decomposition. The method described here is preferable when some of the matrices involved in the computation become ill-conditioned. The inverse of the performance metric g 1  =  JhJT  could be computed by explicitly  inverting h, but problems arise with numerical stability if h is ill-conditioned. This will happen for a manipulator arm if one of the links is much lighter than the others, for example. An alternative method, more suited to this case, is to find a coordinate system on C that brings the metric into Euclidean form (a local Cartesian basis) and evaluate the metric in this frame. In the following it will be shown how this relates to the singular value decomposition. Let a general coordinate transformation in )4) be described by the m x m matrix F  e  GL(m), and one in C by the n x n matrix G E GL(n). GL(n) is the group of  104  Appendix A. Computational Aspects  105  non-singular n x n matrices. The configuration space metric transforms as h  GThG.  —*  Similarly, the work space metric transforms as Ti  FiF,  -4  and the Jacobian transforms as J  —  F’JG.  It follows that 1 g  —*  F_1g_1FT’  which does not depend on G. We can therefore find a G, such that h transforms into the identity matrix. Computing the singular value decomposition [KL8Oj of h, i.e. h=UUT with U orthogonal and E diagonal, gives the desired transformation matrix G as 2 UE”  G  =  =  JUE_ U 1 TJT.  and g’ is given by  A similar method can be used for the generalized Yoshikawa measure (eq. 2.5) and the anisotropy measure (eq. 2.6). Define a matrix A by A  =  g’.  The eigenvalues of A determine the anisotropy measure and the generalized Yoshikawa measure. A is invariant under C pace coordinate transformations. It transforms as A  —*  F’AF  Appendix A. Computational Aspects  106  from which it follows that the eigenvalues do not depend on F. From the singular value decomposition of the work space metric  t,  q=VVT  we obtain a transformation F that brings F  =  in Euclidean form, namely . 2 VE”  we can now express A as A=BBT,  with B  =  The eigenvalues of A can now be computed as the squares of the singular values of B. If one of the matrices (i.e. the Jacobian, the work space and the configuration space metric) is ill-conditioned this will give more accurate results than computing the eigen values of A directly. In this case some of the elements of B will be very small or very large, and computing A will therefore effectively square these entries with a resultant loss of accuracy. However, this method is slower than a straightforward computation of the eigenvalues of A. The above construction could be adapted to a computation of the measures over a path by using a smooth singular value decomposition, [MK89, BGMBN9O]. This will be faster than a recalculation of the singular value decomposition at each point on the path. If all quantities are smooth, the singular value decomposition at nearby points can be computed by small adjustments, in very few iterations, as described in [MK89]. The scalar non-linearity measure as given in eq. 2.9 seems to require the explicit computation of h’ as well as the computation of all the  —  1) components of the  Appendix A. Computational Aspects  107  Riemann curvature tensor, which makes it quite demanding to compute. If there is a more efficient way of computing this quantity seems to be an open problem.  

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