Inverse Kinematics Robert Platt Northeastern University
Inverse Kinematics This addresses the obvious question: what joint angles will place my end effector in a desired pose?
Inverse kinematics Closed form (analytical) solution: a sequence or set of equations that can be solved for the desired joint angles • Potentially faster than an iterative solution • A unique solution to all manipulator positions can be determined a priori . • Can guarantee “safe” joint configurations where the manipulator does not collide with the body. Iterative (numerical) solution: numerical iteration toward a desired goal position (variation on Newton’s method) • Easier to think about • Better suited to incremental displacements and control.
Inverse kinematics There is no general analytical inverse kinematics solution • All analytical inverse kinematics solutions are specific to a robot or class of robots. • based on geometric intuition about the robot • I’ll give one example – there are many variations.
Inverse kinematics q 3 q 4 q 5 q 6 q 2 Spherical wrist: the axes of the last three joints q intersect in a point. 1 Consider this 6-joint robot: • this example is out of the book…
Inverse kinematics q 3 q 4 q 5 Problem: q 6 q • Given: desired transform, 2 • q Find: q q q q q q 1 1 2 3 4 n Note: • The desired transform (pose) encodes six degrees of freedom (this info can be represented by six numbers) • Since we only have six joints at our disposal, there is no manifold of redundant solutions. • For this manipulator, the problem can be decomposed into a position component (the first three joints) and an orientation component (the last three joints) • The first three joints tell you what the position of the spherical wrist
In class exercise Given and , calculate joint angles that cause eff to reach
Example: Inverse kinematics q 3 q 4 q 5 q Solution: 6 q 2 • First, back out the position of the spherical wrist: q 1 Since it’s a spherical wrist, the last three joints can be thought of as rotating about a point. • A constant transform exists that goes from the last wrist joint out to the end sw T effector (sometimes this is called the “tool” transform): eff • Back out the position of the wrist: 1 b b sw T T T sw eff eff
Example: Inverse kinematics q 3 q 4 q 5 q • Next, solve for the first three joints 6 q 2 q 1 q Goal position in horizontal plane First, solve for . (look down from 1 above) 1 q a tan 2 x g y , g or q a tan 2 x g y , 1 g q 1
Example: Inverse kinematics q 3 q 4 q 5 q q 6 q Next, solve for . (look at the 2 3 manipulator orthogonal to the plane of the first two links) q 1 2 2 2 c a b 2 ab cos( ) c 2 2 2 2 r z h l l g g 1 2 cos D c 2 l l q 1 2 3 l 2 2 2 2 r x y l where g g g 1 c h and is the height of the first link q 2 1 D 2 tan q 3 D
Example: Inverse kinematics q 3 q 4 q q 5 Next, solve for . (continue to 2 look at the manipulator q 6 q 2 orthogonal to the plane of the first two links) q 1 z h g tan 2 2 x y g g l s tan 2 3 q l l c 3 1 2 3 l 2 l 1 q 2 q 2
Example: Inverse kinematics Finally, the last three joints completely specify the q orientation of the end effector. 3 q 4 q 5 • Note that the last three joints look just like ZYZ Euler angles q 6 q 2 • Determination of the joint angles is easy – just calculate the ZYZ Euler angles corresponding to the desired orientation. q 1
Remember: ZYZ Euler Angles R zyz ( φ,θ,ψ ) = ( 1 )( cos θ )( 1 ) − sin φ − sin ψ cos φ 0 cos θ 0 sin θ cos ψ 0 sin φ cos φ 0 0 1 0 sin ψ cos ψ 0 0 0 − sin θ 0 0 0 R zyz ( φ,θ,ψ ) = ( c θ ) c φ c θ c ψ − s φ s ψ − c φ c θ s ψ − s φ c ψ c φ s θ s φ c θ c ψ + c φ s ψ − s φ c θ s ψ + c φ c ψ s φ s θ − s θ c ψ s θ s ψ θ =± a tan2 ( √ 1 − r 332 ,r 33 ) a tan 2 r 23 , r k 13 a tan 2 r 32 , r 31
Inverse kinematics for a humanoid arm You can do similar types of things for a humanoid (7-DOF) arm. • Since this is a redundant arm, there are a manifold of solutions… Spherical Spherical shoulder wrist elbow General strategy: 1. Solve for elbow angle 2. Solve for a set of shoulder angles that places the wrist in the right position (note that you have to choose an elbow orbit angle) 3. Solve for the wrist angles
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