Kinematic Models for Robot Compliant Motion with Identiication of Uncertainties

This thesis is about force controlled compliant robot motion, with the emphasis on: 1) modelling of arbitrary and time-varying contact situations between a rigid manipulated object and rigid objects in its environment, 2) motion speci cation in terms of allowed velocities and accelerations for the manipulated object, maintaining the contact with the physical constraints but without generating too large contact forces, and 3) on-line identi cation of uncertainties in the instantaneous geometric parameters of the motion constraint model, i.e., the position of the contact points, the direction of the contact normal, and the local curvature parameters. Requirements for generality, simplicity and robustness have guided the research work. The main new theoretical results of this thesis are: 1) the successful use of the kinematics of mechanisms as the unifying framework for the integration of modelling, speci cation and identi cation; 2) the use of the rst and second order geometric parameters of the contacting surfaces of the manipulated object and its environment as basic model parameters; 3) the derivation of identi cation equations for uncertainties in these parameters, based on the general physical properties of consistency and reciprocity; and 4) new symbolical algorithms to calculate the derivative of the Jacobian matrix of a kinematic chain with respect to one of the chain's joint angles. Real-world application of these theoretical results has been proven feasible by various experiments on a commercial robot arm equipped with force control software of moderate complexity. Beknopte samenvatting Deze thesis handelt over krachtgecontroleerde robots, die taken uitvoeren waarbij contactenmet de omgeving optreden. De nadruk ligt op : 1) het modelleren van willekeurige tijdsvariante contactsituaties waarbij zowel het gemanipuleerde object als de omgeving uit starre lichamen bestaan; 2) het opstellen van een bewegingsspeci catie in termen van snelheden en versnellingen die aan het gemanipuleerde object kunnen opgelegd worden in de gegeven contactsituatie zonder contact te verliezen enerzijds, en anderzijds zonder te grote reactiekrachten op te wekken; en 3) het identi ceren van onzekerheden in het contactmodel, in reele tijd tijdens de taakuitvoering. De vereisten voor algemene geldigheid, eenvoud van concepten, en robuustheid tegen stoorinvloeden vormden steeds de leidraad voor het onderzoek. v vi De belangrijkste theoretische resultaten van deze thesis zijn : 1) de kinematica van mechanismen is op een doorgedreven wijze gebruikt als het uni cerende kader waarin modelleren, speci ceren en identi ceren zijn verenigd; 2) de contactmodellen zijn gebaseerd op de eerste en tweede orde geometrische parameters van de oppervlakken in contact; 3) de begrippen van consistentie en reciprociteit vormen de fysische onderbouw bij het a eiden van de identi catievergelijkingen; en 4) voor de berekening van de afgeleiden van een Jacobiaanmatrix van een kinematische ketting naar een van zijn gewrichtshoeken zijn originele symbolische formules ontwikkeld. De toepasbaarheid van verscheidene van de theoretische resultaten is ook in experimenten met een industriele robot aangetoond. Voor deze experimenten volstaat een relatief eenvoudige robotcontroleprogrammatuur. Symbols and de nitions = : the left-hand side is a shorthand notation for the expression on the righthand side a : scalar (unbold lowercase) a : column vector (bold lowercase) A : matrix (bold uppercase) aold;Jslip : unbold lowercase trailing superscript indicates characteristic feature An m : n-by-m matrix (lowercase unbold subscript), e.g. : I3 3 = 264 1 0 0 0 1 0 0 0 1 375 ; O3 2 = 264 0 0 0 0 0 0 375 ai;Ai : ith vector or matrix in a set (index is unbold lowercase trailing superscript) ai;Ai : ith element of vector a or ith column in matrix A (index is unbold lowercase trailing subscript); e.g.: a = 266664 a1 a2 ... an 377775 ; A = [A1 A2 : : : Am] ax; ay; az : components of three-vector with respect to a reference frame Aij : element of matrix A on row i and column j (index is unbold trailing subscript): An m = 2664 A11 A1m ... . . . ... An1 Anm 3775 aT ;AT : transpose (unbold uppercase T as trailing superscript) vii viii aTb : scalar product AB : matrix product p r : cross product of three-vectors [p ] : matrix representing cross product with vector p: [p ] = 264 0 pz py pz 0 px py px 0 375 Ay : pseudo-inverse of matrix A AyK : weighted pseudo-inverse of matrixA, withK as weighting matrix on the range space of A _ a; _ A : time derivative of a vector a or matrix A e : 6 6 reciprocity matrix: e = " O3 3 I3 3 I3 3 O3 3 # IR : set of real numbers fa; b; cg : set containing elements a, b and c MO : manipulated object (object rigidly connected to robot end e ector) ENV : environment frefg : reference frame (unbold lowercase, between curly braces) fcong : contact frame fprig : principal frame on MO (smooth surface) fpri0g : principal frame on ENV (smooth surface) ffreg : Frenet frame on MO (smooth curve) ffre0g : Frenet frame on ENV (smooth curve) na; refa; oA : indication of reference frame in which scalar, vector or matrix is expressed (unbold leading subscript) p : position vector pn;o : position vector, pointing from n to o X; Y; Z : coordinate axes of right-handed orthogonal reference frame e : unit direction vector: e = 264 ex ey ez 375 eX; eY ; eZ : unit vectors along coordinate axes of a reference frame eX;o; eY;o; eZ;o : unit vectors along coordinate axes of the reference frame fog ix i; j;k : coordinate vectors of unit vectors along coordinate axes i = 264 100 375 ; j = 264 010 375 ; k = 264 001 375 R : revolute joint P : prismatic joint r; r+; r : radius of curvature, radius of maximum curvature, radius of minimum curvature ; +; : curvature = 1=r; + = 1=r+; = 1=r R+; R : revolute joint at radius of maximum, respectively minimum, curvature r0+; r0 ; R0+; R0 : as r+; r ; R+ and R , but for the ENV instead of the MO P+; P ; P 0+; P 0 : prismatic joints along the corresponding radii of curvature ! : angular velocity three-vector: ! = 264 !x !y !z 375 _ ! : angular acceleration three-vector v : translational (or rectilinear) velocity three-vector: v = 264 vx vy vz 375 t : time t : velocity twist of a rigid body or reference frame: t = " !v # tee; tbs : twist of the end e ector frame of a a manipulator with the origin of the end e ector frame, respectively the origin of the base frame, chosen as the reference point for the translational velocity component of the twist eet; bst : twist of the end e ector frame of a a manipulator with its components expressed in the end e ector reference frame, or the base reference frame, respectively. x t : in nitesimal displacement twist (roll x, pitch y, yaw z, translation [dx dy dz]T ): t = 666666664 x y z dx dy dz 777777775 f : force: f = 264 fx fy fz 375 m : moment (torque): m = 264 mx my mz 375 w : wrench (generalized force) on rigid body: w = " f m # tm;wm : measured twist and wrench o nT : 4 4 homogeneous transformation matrix; maps coordinates of threevectors from reference frame fog to reference frame fng: o nT = " o nR pn;o O1 3 1 # : pn;o is the position vector of fog's origin in reference frame fng. o nR : 3 3 rotation matrix from reference frame fog to reference frame fng: o nR = h neX;o neY;o neZ;o i xi o nT : 4 4 in nitesimal homogeneous transformation matrix, corresponding to the in nitesimal displacement twist t : o nT = 26664 1 z y dx z 1 x dy y x 1 dz 0 0 0 1 37775 o nS : 6 6 screw transformation matrix (transformation of twists and wrenches) from frame fog to frame fng: nt = o nS ot; nw = o nS ow; o nS = " o nR O3 3 [pn;o ] o nR o nR # S (t) : 6 6 in nitesimal screw transformation matrix corresponding to the twist t: S (t) = S " !v #! = " [! ] O3 3 [v ] [! ] # o nP : 6 6 screw projection matrix (projection of twist and wrench components) from frame fog to frame fng: nt = o nP ot; nw = o nP ow; o nP = " o nR O3 3 O3 3 o nR # P (t) : 6 6 in nitesimal screw projection matrix corresponding to twist t: P (t) = P " !v #! = " [! ] O3 3 O3 3 [! ] # Mn o : 6 6 reference point transformation matrix (transformation of twist and wrench representations if velocity reference point on the rigid body changes) from origin of frame fog to origin of frame fng: tn =Mn o to nw =Mn o ow Mn o = " I3 3 O3 3 [pno ] I3 3 # xii M (t) : 6 6 in nitesimal reference point transformation matrix corresponding to twist t: M (t) =M " !v #! = " O3 3 O3 3 [v ] O3 3 # q : generalized joint position q; _ q;  q : column vector of generalized joint positions and its derivatives with respect to time: q = 2664 q1 ... qn 3775 ; _ q = 2664 _ q1 ... _ qn 3775 ;  q = 2664  q1 ...  qn 3775 J : twist Jacobian (matrix mapping joint velocities to Cartesian velocities): " !v # = J _ q J+;J : columns of twist Jacobian J corresponding to rotation about R+ and R , respectively J slip : Jslip = h J+ J i J 0+;J 0 : columns of twist Jacobian J corresponding to rotation about R0+ and R0 , respectively J slide : Jslide = h J 0+ J 0 i J r : rotation about contact normal : generalized joint torque ; _ : column vector of generalized joint torques and its time derivative: = 2664 1 ... n 3775 ; _ = 2664 _ 1 ... _ n 3775 G : wrench Jacobian (matrix mapping reaction forces in each contact to resultant Cartesian wrench on the end e ector): " f m # = G xiii = f 1; : : : ; nug : uncertainty parameter set of a kinematic motion constraint model, with nu uncertainty parameters i. Pronunciation: \upsilon" 1 j ; 2 j : jth rst, respectively second, order uncertainty parameter K : 6 6 sti ness matrix C : 6 6 damping matrix I : 6 6 generalized inertia matrix