Lecture Note 7: Velocity Kinematics and Jacobian

Transcription

1 ECE5463: Introduction to Robotics Lecture Note 7: Velocity Kinematics and Jacobian Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 1 / 20

2 Outline Introduction Space and Body Jacobians Kinematic Singularity Outline Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 2 / 20

3 Jacobian Given a multivariable function x = f(θ), where x R m and θ R n. Its Jacobian is defined as [ ] [ ] f J(θ) θ (θ) fi R m n θ j i m,j n If x and θ are both a function of time, then their velocities are related by the Jacobian: [ ] f dθ ẋ = θ (θ) = J(θ) θ dt Let J i (θ) be the ith column of J, then ẋ = J 1 (θ) θ J n (θ) - J i(θ) is the velocity of x due to θ i (while θ j = 0 for all j i) θ n Introduction Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 3 / 20

4 Velocity Kinematics Problem In the previous lecture, we studied the forward kinematics problem, that obtains the mapping from joint angles to {b} frame configuration: θ = (θ 1,..., θ n ) T T sb (θ) In this lecture, we study the velocity kinematics problem, namely, deriving the relation that maps velocities of joint variables θ to the velocity of the end-effector frame - Note: we are interested in relating θ to the velocity of the entire body frame (not just a point on the body) - One may intend to write T (θ) = J(θ) θ. However, T SE(4) and its derivative is not a good way to represent velocity of the body. - Two approaches: (1) Analytical Jacobian: using a minimum set of coordinate x R 6 of the frame configuration and then take derivative; (2) Geometric Jacobian: directly relate θ to the spacial/body twist V Introduction Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 4 / 20

5 Analytical vs. Geometric Jacobian A straightforward way to characterize the velocity kinematics is through the Analytical Jacobian Express the configuration of {b} using a minimum set of coordinates x. For example: - (x 1, x 2, x 3): Cartesian coordinates or spherical coordinate of the origin - (x 4, x 5, x 6): Euler angles or exponential coordinate of the orientation Write down the forward kinematics using the minimum set of coordinate x: x = f(θ) Analytical Jacobian can then be computed as J a (θ) = [ f θ (θ) ] The analytical Jacobian J a depends on the local coordinates system of SE(3) See Textbook for more details Introduction Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 5 / 20

6 Analytical vs. Geometric Jacobian Geometric Jacobian directly finds relation between joint velocities and end-effector twist: [ ] ω V = = J(θ) v θ, where J(θ) R 6 n Note: V = (ω, v) is NOT a derivative of any position variable, i.e. V dx dt (regardless what representation x is used) because the angular velocity is not the derivative of any time varying quantity. Analytical Jacobian J a destroys the natural geometric structure of the rigid body motion; Geometric Jacobian can be used to derive analytical Jacobian. Introduction Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 6 / 20

7 Outline Introduction Space and Body Jacobians Kinematic Singularity Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 7 / 20

8 Geometric Jacobian in Space Form Given the forward kinematics: T sb (θ 1,..., θ n ) = e [S1]θ1 e [Sn]θn M Let V s = (ω s, v s ) be the spacial twist, we aim to find J s (θ) such that V s = J s (θ) θ = J s1 (θ) θ J sn (θ) θ n The ith column J si is the velocity (twist) of the body frame due to only the ith joint motion θ i In other words, J si (θ) is the spatial twist when the robot is rotating about S i at unit speed θ i = 1 while all other joints do not move (i.e. θj = 0 for j i). Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 8 / 20

9 Derivation of Space Jacobian Given any screw axis S, we have d dθ i e [S]θ i = [S]e [S]θ i For simplicity, denote T s,i = e [S 1]θ1 e [S i 1]θ i 1, i = 2,..., n. Let T = T sb (θ) We have [V s] = T T 1 = ( T θ 1 θ1 + + T θ i θi + + T θ n θn ) T 1 Let θ i = 1 and θ j = 0 for j i [J si(θ)] = T θ i T 1 J si(θ) = [ Ad Ts,i ] Si Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 9 / 20

10 Summary of Space Jacobian Given the forward kinematics: T (θ) = e [S1]θ1 e [Sn]θn M Space Jacobian: J s (θ) R 6 n relates joint rate vector θ R n to the spatial twist V s via V s = J s (θ) θ. For i 2, the ith column of J s is J si (θ) = [ Ad Ts,i ] Si, where T s,i = e [S1]θ1 e [Si 1]θi 1 and the first column is J s1 = S 1. Procedure: Suppose the current joint position is θ = (θ 1,..., θ n ): - i = 1: find the screw axis S 1 = (ω s1, v s1) when robot is at home position J s1 = S 1 - i = 2: find the screw axis S 2(θ) = (ω s2, v s2) after moving joint 1 from zero position to θ 1. J s2(θ) = S 2(θ) - i = 3: find the screw axis S 3(θ) = (ω s3, v s3) after moving the first 2 joints from zero position to the θ 1 and θ 2. J s3(θ) = S 3(θ) -... Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 10 / 20

11 Geometric Jacobian Lecture 7 (ECE5463 Sp18) The first joint axis is in the direction Wei Zhang(OSU) ωs1 = (0, 0, 1). 11 / Choo 20 Space Jacobian Example Chapter 5. Velocity Kinematics and Statics θ1 L1 L2 θ3 θ2 θ4 ẑ ŷ ˆx ŷ q3 ẑ ˆx q1 q2 Figure 5.7: Space Jacobian for a spatial RRRP chain. Since the final joint is prismatic, ωs4 = (0, 0, 0), and the joint-axis is given by vs4 = (0, 0, 1). The space Jacobian is therefore Js(θ) = L1s1 L1s1 + L2s L1c1 L1c1 L2c Example 5.3 (Space Jacobian for a spatial RRPRRR chain). We n the space Jacobian for the spatial RRPRRR chain of Figure 5.8. The b is chosen as shown in the figure.

12 Space Jacobian Example (Continued) Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 12 / 20

13 Body Jacobian Recall that V b is the rigid body velocity expressed in body frame. By change of frame for twist, we know V b = [Ad Tbs ] V s or equivalently [V b ] = T bs [V s ]T 1 bs = T 1 sb T sb Body Jacobian J b relates joint rates θ to V b V b = J b (θ) θ = [ J b1 (θ) J bn (θ) ] θ 1. θ n Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 13 / 20

14 Derivation of Body Jacobian Use Body form PoE kinematics formula: T (θ) = Me [B1]θ1 e [Bn]θn For simplicity, denote T b,i+ = e [Bi+1]θi+1 e [Bn]θn, for i = 1,..., n 1 ( ) [V b ] = T 1 T = T 1 T θ θ1 1 + T θ θn n Let θ i = 1 and θ j = 0 for j i, 1 T [J b,i ] = T θ i ) ) = (e [Bn]θn e (e [B1]θ1 [B1]θ1 e [Bi 1]θi 1 [B i ]e [Bi]θi e [Bn]θn = ( T b i+) 1 [Bi ]T b i+ Therefore, J b,i = [ ] Ad T 1 B i b,i+ Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 14 / 20

15 Summary of Body Jacobian Given the forward kinematics: T (θ) = Me [B1]θ1 e [Bn]θn Body Jacobian: J b (θ) R 6 n relates joint rate vector θ R n to the body twist V b via V b = J b (θ) θ. For i = n 1,..., 1, the ith column of J b is [ ] J bi (θ) = Ad T 1 B i, where T b,i+ = e [Si+1]θi+1 e [Sn]θn b,i+ and the last column is J bn = B n. Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 15 / 20

16 Relation Between Space and Body Jacobian Recall that: V b = [Ad Tbs ] V s and V s = [Ad Tsb ] V b Body and spacial twists represent the velocity of the end-effector frame in fixed and body frame The velocity may be caused by one or multiple joint motions. We know the ith column J si (or J bi ) is the spacial twist (or body twist) when θ i = 1 and θ j = 0, j i Therefore, we have J si = [Ad Tsb ] J bi and J bi = [Ad Tbs ] J si. Putting all the columns together leads to J s (θ) = [Ad Tsb ] J b (θ), and J b (θ) = [Ad Tbs ] J s (θ) Detailed derivation can be found in the textbook. Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 16 / 20

17 Outline Introduction Space and Body Jacobians Kinematic Singularity Singularity Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 17 / 20

18 Kinematic Singularity Roughly speaking, kinematic singularity (or simply singularity) refers to a posture at which the robot s end-effector loses the ability to move instantaneously in one or more directions. Mathematically, a singular posture is one in which the Jacobian (J s (θ) or J b (θ)) fails to be of maximal rank Singularity is independent of the choice of space or body Jacobian rankj s (θ) = rankj b (θ) Singularity Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 18 / 20

19 T (θ) =PT Singularity Example Chapter 5. Velocity Kine T (θ) Two collinear revolute joint P (a) Figure 5.10: Kinematic Singularity Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 19 / 20

20 More Discussions Singularity Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 20 / 20