INVERSE KINEMATICS USING ALGEBRAIC METHOD CLOSED FORM SOLUTION

INVERSE KINEMATICS USING ALGEBRAIC METHOD (CLOSED FORM SOLUTION)

Guideline of solving Inverse Kinematics: a)Look at equation involving only one joint variable. Solve this equation first to get the corresponding joint variable solutions. b) Next, look for pairs or set of equations, which could be reduced to one equation in one joint variable by application of algebraic and trigonometric identities. c) Use arc tangent (Atan 2) function instead of arc cosine or arc sine functions. The two argument Atan 2(y, x) function return the accurate angle in the range of –π<=ϴ<=π by examining the sign of both y and x and detecting whenever either x or y is zero. d) Solution in terms of the elements of the position vector components of are more efficient than those in terms of elements of the rotation matrix, as latter may involve solving more complex equations. e)In certain cases, where no direct solution could be found for any of the joint variable, the Inverse Transform approach is used.

Frame {3} L 2 L 1 Frame {2} Frame {0}, {1} i αi-1 ai-1 di ϴi 1 0 0 0 ϴ 1 2 0 L 1 0 ϴ 2 3 0 L 2 0 ϴ 3

INVERSE KINEMATICS USING ALGEBRAIC METHOD (CLOSED FORM SOLUTION) i αi-1 ai-1 di ϴi 1 0 0 0 ϴ 1 2 0 L 1 0 ϴ 2 3 0 L 2 0 0

[ cos(teta 1)*cos(teta 2) - sin(teta 1)*sin(teta 2), - cos(teta 1)*sin(teta 2) - cos(teta 2)*sin(teta 1), 0, L 2*(cos(teta 1)*cos(teta 2) - sin(teta 1)*sin(teta 2)) + L 1*cos(teta 1)] [ cos(teta 1)*sin(teta 2) + cos(teta 2)*sin(teta 1), cos(teta 1)*cos(teta 2) sin(teta 1)*sin(teta 2), 0, L 2*(cos(teta 1)*sin(teta 2) + cos(teta 2)*sin(teta 1)) + L 1*sin(teta 1)] [ 0] 0, 1, [ 1] 0, 0, Transformation matrix to describe frame {3} with respect to frame {0}. Simplify as,

Tdesired is equal to the Transformation matrix, T (frame {3} with respect to frame {0} of the end effector.

CLOSE FORM SOLUTION - STEP 1 Example. Apply guideline (a), Look at equation involving only one joint variable. Solve this equation first to get the corresponding joint variable solutions. You could find -sin ϴ 1 = r 13. However, with guideline (c) this cannot be preferred as correct quadrant of the angle cannot be found. Alternatively, applying guideline (b), ϴ 1 could be isolated by dividing element (2, 1) by (1, 1) or (2, 2) by (1, 2) or (1, 3) by (2, 3) or (2, 4) by (1, 4). Out of these the last one is preferred as per guideline (d).

Applying these guideline to our cases, Guideline (a), none could be found. Guideline (b) and (d) Made use of (3)&(4), square (1) & (2) and add them together, Finally, obtain (6),

If use cos ϴ 2 , quadrant cannot be found. Use guideline (c). From closed form solution where, sin ϴ 2 = a, cos ϴ 2 = b

ϴ 2 is already known, for ϴ 1 , the solution to be in the form of acosϴ - bsinϴ = c asinϴ + bsinϴ = d Where, Therefore, Sign change in the solution of ϴ 2, affect k 2, and thus ϴ 1+ϴ 2+ϴ 3=Atan 2(Sø, Cø)=ø

Summary An inverse kinematic approach is manipulating the given equations into a form for which solution is known.

Trigonometric Equation & Solutions

ANOTHER WAY OF SOLVING INVERSE KINEMATICS (x, y) i αi-1 ai-1 di ϴi 1 0 0 0 ϴ 1 2 0 L 1 0 ϴ 2 3 0 L 2 0 ϴ 3

INVERSE TRANSFORM METHOD

ANOTHER WAY OF SOLVING INVERSE KINEMATICS “INVERSE TRANSFORM METHOD” The left hand side has 3 unknowns i. e. the ϴ 1, ϴ 2, ϴ 3. The left hand side consist of product of n link transformation matrices, i. e. Recall that is a function of only one unknown, Only ϴ 1

ANOTHER WAY OF SOLVING INVERSE KINEMATICS “INVERSE TRANSFORM METHOD” If multiply both sides by the inverse of hand side. Left hand side become, Right hand side become, we can isolate ϴ 1 from left

ANOTHER WAY OF SOLVING INVERSE KINEMATICS Equate (2, 4) elements from both sides of (2) & (3), Left hand side (l. h. s) Right hand side (r. h. s) Complete right hand side multiplication on your own. Focus at element (1, 4) and (2, 4) of l. h. s and the product of r. h. s

ANOTHER WAY OF SOLVING INVERSE KINEMATICS Complete right hand side multiplication on your own. Square both sides, Once C 2 is solved, then proceed with C 1.

ASSIGMENT 2 q. Determine the DH parameters. q. Calculate the transformation matrix from Tool Point to Base (Link 0). q. Obtain the inverse kinematics solutions using both methods closed form and inverse transform approach.