Lecture Note 7: Velocity Kinematics and Jacobian
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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 / 19
2 Outline Introduction Space and Body Jacobians Kinematic Singularity Outline Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 2 / 19
3 Jacobian Given a multivariable function x = f(θ), where x R m and θ R n.its Jacobian is defined as J(θ) [ ] f θ (θ) [ ] fi θ j i m,j n R m n If x and θ are both a function of time, then their velocities are related by the Jacobian: ẋ = [ ] f dθ θ (θ) dt = J(θ) θ Let J i (θ) be the ith column of J, thenẋ = J 1 (θ) θ J n (θ) θ n - J i (θ) is the velocity of x due to θ i (while θ j =0for all j i) Introduction Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 3 / 19
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 / 19
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 / 19
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 / 19
7 Outline Introduction Space and Body Jacobians Kinematic Singularity Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 7 / 19
8 Geometric Jacobian in Space Form Given the forward kinematics: T sb (θ 1,...,θ n )=e [S 1]θ1 e [S n]θ n 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 =1while all other joints do not move (i.e. θj =0for j i). Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 8 / 19
9 Derivation of Space Jacobian Given any screw axis S, wehave d dθ i [S]θ i =[S]e [S]θ i e For simplicity, denote T s,i = e [S 1]θ1 e [S i 1]θ i 1, i =2,...,n.LetT = T sb (θ) We have [V s ]= TT 1 = ( ) T θ1 θ T θi θ i + + T θn θ n T 1 Let θ i =1and θ j =0for j i [J si (θ)] = T θ i T 1 [ ] = Ad Ts,i Si Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 9 / 19
10 Summary of Space Jacobian Given the forward kinematics: T (θ) =e [S 1]θ1 e [S n]θ n 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. Fori 2, theith column of J s is J si (θ) = [ Ad Ts,i ] Si, where T s,i = e [S 1]θ1 e [S i 1]θ i 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 / 19
11 Space Jacobian Example θ 1 L 1 L 2 θ 3 θ 2 θ 4 ẑ ŷ ˆx ŷ q 3 ẑ ˆx q 1 q 2 Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 11 / 19
12 Space Jacobian Example (Continued) Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 12 / 19
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 / 19
14 Derivation of Body Jacobian Use Body form PoE kinematics formula: T (θ) =Me [B 1]θ1 e [B n]θ n For simplicity, denote T b,i+ = e [B i+1]θ i+1 e [B n]θ n,fori =1,...,n 1 ( ) [V b ]=T 1 T = T 1 T θ θ1 1 + T θ θn n Let θ i =1and θ j =0for j i, 1 T [J b,i ]=T θ i ) = (e [B n]θn e [B 1]θ 1 )(e [B 1]θ1 e [B i 1]θ i 1 [B i ]e [B i]θi e [B n]θ n = ( 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 / 19
15 Summary of Body Jacobian Given the forward kinematics: T (θ) =Me [B 1]θ1 M e [B n]θ n 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. Fori = n 1,...,1, theith column of J b is [ ] J bi (θ) = Ad T 1 S, where T b,i+ = e [S i+1]θ i+1 e [S n]θ n i b,i+ and the last column is J bn = B n. Geometric Jacobian Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 15 / 19
16 Relation Between Spatial 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 =1and 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 / 19
17 Outline Introduction Space and Body Jacobians Kinematic Singularity Singularity Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 17 / 19
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 / 19
19 Singularity Example Two collinear revolute joint Singularity Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 19 / 20
20 More Discussions Singularity Lecture 7 (ECE5463 Sp18) Wei Zhang(OSU) 20 / 19
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