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 R d 6 q eff eff • = Given: desired transform, T 2 eff 0 1 ( ) • 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
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 effector (sometimes this is called the “tool” transform): sw T 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 ( ) = + π 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 − θ c a b 2 ab cos( ) c ( ) 2 2 2 2 + − − − r z h l l ( ) g g 1 2 cos θ = − = − D c 2 l l q 1 2 3 l 2 2 2 2 = + r x y l where θ 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 θ = tan 2 2 + x y g g l s ( ) α = tan 2 3 q + l l c 3 1 2 3 l 2 l 1 = θ ± α q α 2 q θ 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 q 6 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 φ − φ θ θ ψ − ψ cos sin 0 cos 0 sin cos sin 0 ( ) φ θ ψ = φ φ ψ ψ R , , sin cos 0 0 1 0 sin cos 0 zyz − θ θ 0 0 1 sin 0 cos 0 0 1 − − − c c c s s c c s s c c s φ θ ψ φ ψ φ θ ψ φ ψ φ θ ( ) φ θ ψ = + − + R zyz , , s c c c s s c s c c s s φ θ ψ φ ψ φ θ ψ φ ψ φ θ − s c s s c θ ψ θ ψ θ 2 θ = ± − a tan 2 1 r 33 , r 33 ( ) φ = + π a tan 2 r 23 , r k 13 ( ) ψ = 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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