Inverse ManipulatorKinematicsAlgebraic solution by reduction to polynomialOutline2 Introduction IntroductionIntroductionThe Inverse kinematic is the basis of robot trajectory planning and control.5IntroductionExample :6Algebraic solution by reduction to polynomialOutline7SolvabilitySolvabilityFor the 6 DOF Puma 560 manipulator,we have:How to find the 6 joint variablesHere we might have 12 equations to solve for 6 independent variables. Constraints should be utilized.6 equations for 6 unknown variables9SolvabilityDifficulty: these 6 equations are nonlinear and transcendental equations.obtain the solution.whereSolvability11SolvabilitySolvabilityThe dexterous workspace is only one point(the origin). The There is no dexterous workspace. The reachable SolvabilityFor most industry robots, there is limitation for the joint variable range, thus the workspace is reduced.Only one attainable orientationIf a manipulator has less than 6 DOF, it can’t attain general goal position and orientation in 3D space.Workspace also depends on the tool-frame transformation.Solvability15There might be multiple solution in solving kinematic equations.Two possible solution for the same position and orientation.How to choose possible solution?Solvability” solution.The number of solutions depends on the number of and the allowable ranges of motion of the joints, also, it can be a function of other link parameters (link length, link twist, link offset, joint angle).Solvability2. Multiple solutions17The PUMA 560 can reach certain goals with 8different solutions.+Due to the limits of joints range, some of these 8 solutions could be inaccessible.SolvabilitySolvabilityAlgebraic solution by reduction to polynomial Outline20Manipulator Subspace21workspace is a portion of an n‐DOF subspacesubspace : planeworkspace : a subset of the plane{workspace} ⊂{subspace} ⊂{space}Manipulator Subspaceof a manipulator?Giving an expression for a manipulator’s wrist frame {w}to be free to take on all possible values.Manipulator SubspaceThe subspace of is given by:233R planar manipulatorAs are allowed to take on arbitrary values, the subspace is generatedNOTE : Link lengths and joints limits restrict the workspace of the manipulator to be a subset of this subspace.Algebraic solution by reduction to polynomial Outline24Algebraic vs. GeometricGiven the transformation matrix, solved for25Algebraic vs. GeometricD-H TableAlgebraic vs. GeometricThe transformation matrix can be computed viaand we haveAlgebraic vs. GeometricSpecification of the goal points can be accomplished by specifying three parameters: ..The transformation is assumed to have the following structurewhereThe above four nonlinear equations are used to solve for (unknown)Algebraic vs. GeometricThe parameters is How to solve for according thefollowing equations:Algebraic vs. Geometric1.Algebraic solution 30The is the only unknown parameter.Algebraic vs. GeometricStep1.In the solution algorithm, the above constraintshould be checked to determine whether a solution exist or not. If the constrain is not Algebraic vs. Geometric1.Algebraic solution Here, the choice of signs in the solution of corresponds to Algebraic vs. Geometric33Based on the solution of , we can get:whereAlgebraic vs. Geometricwe haveAlgebraic vs. GeometricNote:If a choice of sign is made in the solution of ,it will affect and thus affectStep5. Based on the fact that The solution of can be obtained.Algebraic vs. Geometric36solved for by using the tools of plane geometry.can utilize plane geometry directly to find a solution.Algebraic vs. Geometricconsidering the solid triangle, the “” can be applied to solve for as:37PossibleconfigurationThe other possible solution can be obtained by settingAlgebraic vs. Geometric2. Geometric solutionTo solve for , we find the express for angleand .38and can be solved via:then can be solved as:Algebraic vs. Geometric39the solution of can Algebraic solution by reduction to polynomial Outline40Algebraic solution by reduction to polynomialexpression in terms of a single variable.This is a very important geometric substitution used often in solving kinematic equations. These substitution convert transcendental equations into polynomial equations in Algebraic solution by reduction to polynomialGiven a transcendental equation try to solve for42Solutions:(when )Algebraic solution by reduction to polynomial Outline43Inverse manipulator kinematicsThe Unimation Puma 560 Industry Robot44Inverse manipulator kinematicsReview : D-H table45Inverse manipulator kinematicsReview : Transformation of each link.46Inverse manipulator kinematicsReview : Transformation of all link47whereInverse manipulator kinematics: Given the goal point and orientation specified by:(Known: Numerical value)Solve forInverse manipulator kinematics Separating out 1 unknown parameter How to solve ?Inverse manipulator kinematics2. Inverting to be obtain50 whereInverse manipulator kinematicsCheck the (2,4) elements on both sides ,we have Inverse manipulator kinematicsIntroduce the trigonometric(三角恒等变换) substitutions:52whereThen it can be obtained that:Inverse manipulator kinematics3. The left side of the following equation is known53Inverse manipulator kinematicsTaking square of the above two equations, and adding the results together, it can be obtained thatInverse manipulator kinematicsThe above equation depends only on , then similar steps can be followed to solve for as:4. Consider the following equationhave been solved, but is unknownInverse manipulator kinematics56Eq.(3.11) in Chapter3Check elements (1,4) and (2,4) on both sides, we haveInverse manipulator kinematics 57Inverse manipulator kinematics585. Now the left side of the following equation is knownEq.(3.11) in Chapter3Check the elements (1,3) and (3,3), it can be obtained thatInverse manipulator kinematics ca can be solved as:Case2.,The manipulator is in a singular configurationas axis 4 and 6 line up and cause the same motion of the last link of the robot. Thus is chosen arbitrarily.Inverse manipulator kinematics606. Consider the following equation again:andCheck the elements (1,3) and (3,3), it can be obtained thatInverse manipulator kinematics 61Hence, we can solve for as7. Applying the same method one more time, we havewhereCheck the elements (3,1) and (1,1), it can be obtained thatInverse manipulator kinematics62Thus we can solve for aswe can obtain eight sets of possible solutions, some of them will be discarded due to the joint angle limitsInverse manipulator kinematics63Summary1、原则：等号两端的矩阵中对应元素相等，列出相关方）、从含变量少的左边开始，如，向右递推，直到）、选择等号左边或右边矩阵中等于常数或仅含有一个变量的元素，列出相应元素对应的方程或方程组。