MECH 498: Introduction to Robotics Inverse Manipulator Kinematics M. O’Malley
Manipu Manipulato latorr Kinemati Kinematics cs – Example Example – 3R
Direct vs. Inverse Kinematics •
Direct (Forward) Kinematics – Given: Given: Join Jointt angles angles and link links s geometry – Comput Compute: e: Positi Position on and orientation of the end effector relative to the base frame
•
Inverse Kinematics – Given: Given: Posit Position ion and and orienta orientatio tion n of the end effector relative to the base frame – Comput Compute: e: All All possib possible le sets sets of joint angles and link geometries which could be used to attain the given position and orientation of the end effector
Solv Solvabi abilility ty – PUMA PUMA 560 560
Solvability • Existence of Solutions • Multiple Solutions • Method of solutions – Closed Closed form form solution solution • Algebr Algebraic aic soluti solution on • Geome Geometri tric c solu solutio tion n
– Numeri Numerical cal solutions solutions
Solvab Solvabili ility ty – Existen Existence ce of Solution • For a solutio solution n to exist, exist, must must be in the workspace of the manipulator • Workspace - Defi Defini niti tion ons s – Dexterous Workspace (DW): (DW): The subset of space in which the robot end effector can reach in all orientations. orientations . – Reachable Workspace (RW): The subset of space in which the robot end effector can reach in at least 1 orientation
• The Dexter Dexterous ous Workspa Workspace ce is is a subse subsett of the Reachable Workspace
Solvab Solvabili ility ty - Existe Existence nce of of Soluti Solution on - Worksp Workspace ace - 2R Example 1 – L1 = L2
Solvab Solvabili ility ty - Existe Existence nce of of Soluti Solution on - Worksp Workspace ace - 2R Example 2 – L1 ≠ L2
Solvab Solvabili ility ty - Existe Existence nce of of Soluti Solution on - Worksp Workspace ace - 3R Example 3 – L1 = L2
Solvabi Solvabilit lity y – Multipl Multiple e Solutio Solutions ns • Mult Multip iple le sol solut utio ions ns are are a common problem that can occur when solving inverse kinematics because the system has to be able to chose one • The The num numbe berr of of solu soluti tion ons s depends on the number of joints in the manipulator but is also a function of the link parameters (ai , α i , θ i , d ) i • Exam Exampl ple: e: The The PUM PUMA A 560 560 can reach certain goals with 8 different arm configurations (solutions)
Solvabi Solvabilit lity y – Multipl Multiple e Solutio Solutions ns • Problem: The fact that a manipulator has multiple solutions may cause problems because the system has to be able to choose one • Solution: Decision criteria – The closes closestt (geometr (geometrica ically lly)) • Minimi Minimizin zing g the amoun amountt that that each each joint is required to move • Note Note 1: 1: inpu inputt argu argumen mentt - presen presentt position of the manipulator • Note Note 2: Joint Joint Weig Weight ht - Moving Moving small small joints (wrist) instead of moving large joints (Shoulder & Elbow)
– Obstacles Obstacles exist in the the workspace workspace • Avoi Avoidi ding ng coll collis isio ion n
Solvabil Solvability ity – Multiple Multiple Solutions Solutions – Number of Solutions • Task Task Defi Defini niti tion on - Posi Positi tion on the the end effector in a specific point in the plane (2D) • No. No. of of DOF DOF = No. No. of DOF DOF of of the the task – Numbe Numberr of of solut solution ions: s: • 2 (elb (elbow ow up/d up/dow own) n)
• No. No. of of DOF DOF > No. No. of DOF DOF of of the the task – Numbe Numberr of of solut solution ions: s:
∞
– Self Self Moti Motion on - The robot robot can be moved without moving the end effector from the goal
Solvab Solvabili ility ty – Methods Methods of Solut Solution ions s • Solution (Inverse (Inverse Kinematics Kinematics))- A “solution” “solution” is the set of joint variables associated with an end effector’s desired position and orientation. • No general algorithms that lead to the solution of inverse kinematic kinematic equations. equations. • Solution Strategies – Closed form Solutions - An analyti analytic c expressi expression on includes all solution sets. • Algebraic Solution - Trigonometric (Nonlinear) equations • Geometric Solution - Reduces Reduces the larger larger problem problem to a series of plane geometry problems.
– Numerical Solutions - Iterative Iterative solution solutions s will not be be considered in this course.
Solvability
Mathematical Equations • Law Law of of Sine Sines s / Cos Cosin ines es - For For a gen gener eral al tri trian angl gle e
• Sum of Angles
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
Invers Inverse e Kinemat Kinematics ics - Planar Planar RRR RRR (3R) (3R) - Algebr Algebraic aic Solution Solution
where
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
c22 + s22 = 1
Note: The choice of the sign corresponds to the multiple solutions in which we can choose the “elbow-up” or the “elbow-down” solution
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution Based on the previous two transformations, the equations can be rewritten as:
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
Inverse Inverse Kinematic Kinematics s - Planar Planar RRR (3R) (3R) Algebraic Solution
Based on the original equations,
Central Central Topic Topic – Inverse Inverse Manipu Manipulato lator r Kinem Kinemati atics cs - Exampl Examples es • Geometri Geometric c Solution Solution – Concept Concept – Decompos Decompose e spatial spatial geomet geometry ry into several several plane geometry problems – Examples - Planar RRR (3R) manipulators - Geometric Solution
• Algebrai Algebraic c Solution Solution - Concept Concept
– Examples - PUMA PUMA 560 560 - Algebr Algebraic aic Solution
Inverse Inverse Kinema Kinematics tics - Planar Planar RRR (3R) (3R) Geometric Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Geometric Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Geometric Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Geometric Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Geometric Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
,
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
The solution for θ1
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution
Invers Inverse e Kinema Kinematics tics - PUMA PUMA 560 Algebraic Solution • After After all all eigh eightt soluti solution ons s have have been been computed, some or all of them may have to be discarded because of joint limit violations. • Of the the remai remaining ning valid valid soluti solutions, ons, usually usually the one closest to the present manipulator configuration is chosen.
Centra Centrall Topic Topic - Invers Inverse e Manipul Manipulato ator r Kinematics -Examples • Geometric Solution Concept – Decomp Decompose ose spatia spatiall geometry into several plane geometry – Example - 3D - RRR RRR (3R) (3R) manipulat manipulators ors - Geometric Geometric Solution
• Algebraic Solution (closed form) – – Piep iepers ers Meth Method od - Last three consecutive axes intersect at one point – Example - Puma Puma 560 560
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le By applying the law of cosines, we get
θ 3 = Atan2( Atan2(
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le
β
α
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le
Algebraic Solution by Reduction to Polynomial • Transcend Transcendental ental equations equations are difficult difficult to solve solve because because they are a function of c of c θ θ, sθ
• Making Making the the following following substitution substitutions s yields yields an expression expression in terms of a single variable u • Using Using this subst substitu itutio tion, n, transce transcende ndenta ntall equation equations s are converted into polynomial equations
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le
Algebraic Solution by Reduction to Polyn Polynom omia iall - Examp Example le
b2 + a2 – c2
b2 + a2 – c2
Solvability
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect • Pieper’s Solution – Closed Closed form form solutio solution n for for a seria seriall 6 DOF in which three consecutive axes intersect at a point (including robots with three consecutive parallel axes, since they meet at a point at infinity)
• Piep Pieper er’s ’s met metho hod d appl applie ies s to the the majority of commercially available industrial robots – Exampl Example: e: (Puma (Puma 560) 560) • All 6 joint joints s are revol revolute ute joint joints s • The last last 3 joints joints are intersect intersecting ing
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect • Given: – Manipulator Geometry: 6 DOF & DH parameters • All 6 joints joints are revolute revolute joints joints • The last last 3 joints joints are are interse intersecting cting
– Goal Point Definition: Definition: The position and orientation of the wrist in space
• Problem: – What are the joint angles angles ( θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6 ) as a function of the goal (wrist position and orientation)
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect • When the last last three three axes of a 6 DOF DOF robot robot inters intersect, ect, the origins of link frame {4}, {5}, and {6} are all located at the point of intersection. This point is given in the base coordinate system as • From From the gene general ral forw forward ard kine kinemat matics ics meth method od for for determining homogeneous transforms using DH parameters, we know:
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
variable is zero
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Pieper Pieper’s ’s Solu Solution tion - Three Three consecutive Axes Intersect
Centra Centrall Topic Topic - Invers Inverse e Manipul Manipulato ator r Kinem Kinemat atics ics - Exam Exampl ples es • Algebraic Solution (closed form) – – Piepe ieperrs Meth Method od (Continued) • Last Last three three consec consecuti utive ve axes axes intersect at one point
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist • Conside Considerr a 3 DOF DOF non-p non-plan lanar ar robot robot whose whose axes axes all intersect at a point.
Mapp Mappin ing g - Rota Rotate ted d Fram Frames es - Z-YZ-Y-X X Euler Angles
Mapp Mappin ing g - Rota Rotate ted d Fra Frame mes s - ZYX ZYX Euler Angles
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist • Because, Because, in this this example, example, our robot robot can can perform no translations, we can write
• The above above transf transform orm provides provides the solution solution to the forward kinematics.
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist • The The inver inverse se kine kinema mati tics cs probl problem em.. – Given Given a particu particular lar rotatio rotation n - Goal (again, (again, this this robot robot can perform no translations) – Solve: Solve: Find the the Z-Y-X Z-Y-X Euler Euler angles angles
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist
Three Three consecutive consecutive Axes Axes Intersect Intersect - wrist wrist • Unfo Unfort rtun unat atel ely, y, whi while le this this seem seems s like a simple solution, it is troublesome in practice because α is never exactly zero. This leads to singularity problems • For For this this exa examp mple le,, the the sing singul ular ar case results in the capability for self-rotation. That is, the middle link can rotate while the end effector’s orientation never changes.