Introduction Robotics Lecture - Day 4 Out Of 7.ppt
1/45Introduction Roboticsdr Dragan KostićWTB Dynamics and ControlSeptember - October 2009Introduction Robotics, lecture 4 of 7
2/45Outline Recapitulation Velocity kinematics Manipulator Jacobian Kinematic singularities Inverse velocity kinematicsIntroduction Robotics, lecture 4 of 7
3/45RecapitulationIntroduction Robotics, lecture 4 of 7
4/45The general IK problem (1/2) Given a homogenous transformation matrix H SE(3)find (multiple) solution(s) q1, ,qn to equation Here, H represents the desired position and orientation of the tipcoordinate frame onxnynzn relative to coordinate frame o0x0y0z0of the base; T0n is product of homogenous transformationmatrices relating successive coordinate frames:Introduction Robotics, lecture 4 of 7
5/45The general IK problem (2/2) Since the bottom rows of both T0n and H are equal to [0 0 0 1],equationgives rise to 4 trivial equations and 12 equations in n unknownsq1, ,qn:Here, Tij and Hij are nontrivial elements of T0n and H.Introduction Robotics, lecture 4 of 7
6/45Kinematic decoupling (1/3) General IK problem is difficult BUT for manipulators having 6 jointswith the last 3 joint axes intersecting at one point, it is possible todecouple the general IK problem into two simpler problems:inverse position kinematics and inverse orientation kinematics. IK problem: for given R and o solve 9 rotational and 3 positionalequations:Introduction Robotics, lecture 4 of 7
7/45Kinematic decoupling (2/3) Spherical wrist as paradigm. Let oc be the intersection of the last 3 joint axes; as z3, z4, and z5intersect at oc, the origins o4 and o5 will always be at oc;the motion of joints 4, 5 and 6 will not change the position of oc;only motions of joints 1, 2 and 3 can influence position of oc.Introduction Robotics, lecture 4 of 7
8/45Kinematic decoupling (3/3) q1, q2, q3 Introduction Robotics, lecture 4 of 7 q4, q5, q6
9/45Velocity KinematicsIntroduction Robotics, lecture 4 of 7
10/45Scope Mathematically, forward kinematics defines a function betweenthe space of joint positions and the space of Cartesian positionsand orientations of a robot tip; the velocity kinematics are thendetermined by the Jacobian of this function. Jacobian is encountered in many aspects of robotic manipulation:in the planning and execution of robot trajectories, in thederivation of the dynamic equations of motion, etc.Introduction Robotics, lecture 4 of 7
11/45Angular velocity: the fixed axis case When a rigid body moves in a pure rotation about a fixed axis,every point of the body moves in a circle; the centers of all thesecircles lie on the axis of rotation. Let θ be the angle swept out by the perpendicular from a point tothe axis of rotation; if k is a unit vector in the direction of the axisof rotation, then the angular velocity is given by Given the angular velocity ω, the linear velocity of any point iswhere r is a vector from the origin (laying on the axis of rotation)to the point.Introduction Robotics, lecture 4 of 7
12/45Skew symmetric matrices An n n matrix S is skew symmetric if and only if The set of all such matrices is denoted by so(n). From this definition, we see that the diagonal elements of Sare zero, i.e. sii 0; also, we see that S so(3) containsonly 3 independent entries and has the formIntroduction Robotics, lecture 4 of 7
13/45Properties of skew symmetric matrices For a vector a [ax, ay, az]T we defineIntroduction Robotics, lecture 4 of 7
14/45The derivative of a rotation matrix If R(θ) SO(3), then R(θ)RT (θ) I. Differentiating both sidesw.r.t. θ yields Multiplying both sides on the right by R and using the factthat ST -S, we getdR(θ ) R T (θ ) R(θ ) SR(θ ) 0.dθ Since R(θ)RT (θ) I , we obtain:[Introduction Robotics, lecture 4 of 7]
15/45Derivative of Rx,θ as an exampleHence:Similarly we can getIntroduction Robotics, lecture 4 of 7
16/45Derivative of Rl,θ Let Rl,θ be a rotation matrix about the axis defined by unit vector l.ThenIntroduction Robotics, lecture 4 of 7
17/45Angular velocity: general case Consider angular velocity ω about an arbitrary, possibly moving,axis. Suppose that R(t) SO(3) is a time-dependent rotationmatrix. Thenwhere ω(t) is the angular velocity of the rotating frame with respectto the fixed frame at time t.Introduction Robotics, lecture 4 of 7
18/45Proof that ω is the angular velocity vector If p is a point rigidly attached to a moving frame, thenDifferentiating, we obtainIntroduction Robotics, lecture 4 of 7
19/45Addition of angular velocities (1/3) Let o0x0y0z0, o1x1y1z1, and o2x2y2z2 be three frames such that- o0x0y0z0 is fixed,- all three share a common origin,- R01(t) and R12(t) represent time-varying relative orientationsof frames o1x1y1z1 and o2x2y2z2. Also let ωki,j denotes the angular velocity vector correspondingto the derivative of Rij, expressed relative to the frame okxkykzk.Introduction Robotics, lecture 4 of 7
20/45Addition of angular velocities (2/3)Introduction Robotics, lecture 4 of 7
21/45Addition of angular velocities (3/3) For an arbitrary number of coordinate systems:Introduction Robotics, lecture 4 of 7
22/45Linear velocity of a point attached to a moving frame (1/2) Suppose that p is rigidly attached to the frame o1x1y1z1 and thato1x1y1z1 is rotating relative to the frame o0x0y0z0. Then, we haveIntroduction Robotics, lecture 4 of 7
23/45Linear velocity of a point attached to a moving frame (2/2) Suppose that the motion of o1x1y1z1 relative to o0x0y0z0 is given bya homogeneous transformation matrix Dropping the argument t, subscripts and superscripts, we getwhere r Rp1 (vector from o1 to p expressed in the orientation ofo0x0y0z0) and υ is the velocity at which the origin o1 is moving.Introduction Robotics, lecture 4 of 7
24/45Manipulator JacobianIntroduction Robotics, lecture 4 of 7
25/45Derivation of the Jacobian Consider an n-link manipulator with joint variables q1, q2, , qn. Let q [q1, q2, , qn]T. Let the transformation from the end-effector to thebase frame be: Let the angular velocity of the end-effector ω0n beKarl Gustav Linear velocity of the end-effector isJacob Jacobi We seek expressions(1804-1851)Introduction Robotics, lecture 4 of 7
26/45The manipulator Jacobian The manipulator Jacobian:Karl GustavJacob Jacobi(1804-1851)Introduction Robotics, lecture 4 of 7
27/45Angular velocity If the ith joint is revolute: the axis of rotation is given by zi 1;let ωi 1i 1,i represent the angular velocity of the link i w.r.t. theframe oi 1xi 1yi 1zi 1. Then, we have If the ith joint is prismatic: the motion of frame i relative to framei-1 is a translation andIntroduction Robotics, lecture 4 of 7
28/45Overall angular velocity By using already derived formulawe getω00,n ρ1q&1 z00 ρ 2 q& 2 R10 z11 K ρ n q& n Rn0 1 z nn 11 ρ1q&1 z00 ρ 2 q& 2 z10 K ρ n q& n z n0 1 ,whereIntroduction Robotics, lecture 4 of 7
29/45Angular velocity Jacobian The complete Jacobian: Jacobian for angular velocities:Introduction Robotics, lecture 4 of 7
30/45Linear velocity Jacobian The linear velocity of the end effector is just By the chain rule for differentiationwe find Jacobian for linear velocitiesIntroduction Robotics, lecture 4 of 7
31/45Case 1: prismatic jointsIntroduction Robotics, lecture 4 of 7
32/45Case 2: revolute joints The linear velocity of theend-effector is of formwhere Hence we getIntroduction Robotics, lecture 4 of 7
33/45Combining the linear and angular velocity Jacobians The Jacobian is given bywhereandIntroduction Robotics, lecture 4 of 7
34/45Computation of the Jacobian We need to compute The former is equal to the first three elements of the 3rd columnof matrix T0i, whereas the latter is equal to the first threeelements of the 4th column of the same matrix. Conclusion: it is straightforward to compute the Jacobianonce the forward kinematics is worked out.Introduction Robotics, lecture 4 of 7
35/45Kinematic singularitiesIntroduction Robotics, lecture 4 of 7
36/45Kinematic singularities The 6 n manipulator Jacobian J(q) defines mappingξ J (q )q& All possible end-effector velocities are linear combinations ofthe columns Ji of the Jacobianξ J1q&1 J 2 q& 2 K J n q& n The rank of a matrix is the number of linearly independentcolumns (or rows) in the matrix; for J RR6 n:rank J min(6, n) The rank of Jacobian depends on the configuration q; at singularconfigurations, rankJ(q) is less than its maximum value.Introduction Robotics, lecture 4 of 7
37/45Properties of kinematic singularities At singular configurations:– certain directions of end-effector motion may be unattainable,– bounded end-effector velocities may correspond to unboundedjoint velocities,– bounded joint torques may correspond to unboundedend-effector forces and torques. Singularities correspond to points:– on the boundary of the manipulator workspace,– within the manipulator workspace that may be unreachable under smallperturbations of the link parameters (e.g. length, offset, etc.).Introduction Robotics, lecture 4 of 7
38/45Examples of kinematic singularities (1/2)Introduction Robotics, lecture 4 of 7
39/45Examples of kinematic singularities (2/2)Introduction Robotics, lecture 4 of 7
40/45Inverse velocity kinematicsIntroduction Robotics, lecture 4 of 7
41/45Inverse velocity problem The Jacobian kinematic relationship: The inverse velocity problem is to find joint velocities that producethe desired end-effector velocity. When Jacobian is square (manipulator has 6 joints) and nonsingular,one gets: If the number of joints is not exactly 6, J cannot be inverted; then theinverse velocity problem has a solution (obtained using e.g. Gaussianelimination) if and only ifIntroduction Robotics, lecture 4 of 7
42/45Pseudoinverse of Jacobian When number of joints n is above 6, the manipulator iskinematically redundant; then, the inverse velocity problemcan be solved using the pseudoinverse of J. Suppose that rank J m and m n. Then, the right pseudoinverseof J is given by Note that It holdswhere b RRn is an arbitrary vector.Introduction Robotics, lecture 4 of 7
43/45Computation of pseudoinverse Take the singular value decomposition of J aswhere U RRm m and V RRn n are both orthogonal matrices andΣ RRm n is given bymIntroduction Robotics, lecture 4 of 7
44/45Formula for pseudoinverse The right pseudoinverse of J iswhereTmIntroduction Robotics, lecture 4 of 7
45/45Measures of kinematic manipulability Indicate how close is manipulator to a singular configuration. In terms of singular values σi of the manipulator Jacobian J,kinematic manipulability is defined by:µ σ1 σ 2 L σ m In terms of eigenvalues λi of J or determinant of J, µ is given by:Tµ det JJ λ1 λ2 L λm Condition number of J is another manipulability measure:max σ icond J ; i 1,L, m.Introduction Robotics, lecture 4 of 7min σ i
Introduction Robotics, lecture 4 of 7 J q& J q& K Jnq& n ξ 1 1 2 2 The rank of a matrix is the number of linearly independent columns (or rows) in the matrix; for J RRRR6 n: rank J min( 6, n) The rank of Jacobian depends on the configuration q; at singular configurations , rank J(q) is less than its maximum value.
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