Inverse Kinematics Find the required joint angles to place the robot at a given location Places the frame {T} at a point relative to the frame {S} Often Can be split into two parts –First find the {W}using transformations relative to {B} –Then find the inverses to find the joint angles In other words Given Find The problem of solving these equation in non-linear –Puma can reach a position with up to 8 different joint solution. –In general a 6DOF manipulator made of rotary actuator could have 16 solutions. –The more zero link lengths and the more   equal to the simpler the solution

Kinematics Preliminaries Relation Ship between adjacent links can be described by a transformation matrix as Shown. A six DOF manipulator could be described by 7 matrices Vectors In R Describe Orientation P describes Position 16 Values –4 Trivial –3 for position –9 for orientation only 3 unique

Loop Closure Equation In general all unknowns are on Right –Knowns on Left –Unknowns multiply each other and are complex. can be rewriten in many way –Piepers –etc Some forms will be simpler. –Find Inverse of Wrist –Wrist Point Robot

Several Methods can be used Suppressed Tangent Method Suppressed Joint Variables Long Form

Number of Solutions a 1 =a 2 =a 5 =0<4 a 3 =a 5 =0<8 a 3 =0<16 no a i =0<16 If any 3 neighboring joints intersect then there will be a solution. Spherical Joint XYZ Etc.

Definition Linear: For each x there is uniquely one y. Additive and Multiplicative, F(2x)=F(x)+F(x). Solvable: Has a unique solution Closed Form: Can be computed Algebraically and provides all solutions Numerical: Iterative Process Algebraic: Uses Algebraic Identities to substitute and to solve Geometric: Utilizes Geometric Relationships Law of Sines, etc Sub-Space: Smaller Region Circle in a plane Sphere in XYZ space

Main Methods Numerical –Iterative Closed Form –Algebraic –Geometric All 6DOF manipulators in a single serial chain are solvable numerically

Mobility Plane System loses a degree of freedom for each constraint Point has 2 DOF –So n points having C constraints has f=2n-c DOF –2 points constrained by a link would have 3 DOF Rigid bodies thus has 3DOF in a plane –So N rigid bodies in a plane have 3N DOF –If they are connected by g joints each having u i constraints – Or in terms of freedoms –Or –For an open serial Chain N=g –For a closed Chain g=N+1 Since the background is a link and –Has no DOFs 2 DOF n=4 g=4 F=1 3 DOF n=2 C=1 n=4 g=4 F=3

Mobility Space Point had 3DOF Rigid Body is 3 Points Connected by Constraints –F=3n-CF=9-3=6 –Rigid Body has 6 DOF So for a system of bodies Counting Ground as a Body and writing in terms of freedoms For Open Loop chain Or for Closed Loop have one more joint than links

Inverse Kinematics Heuristic 1.Equate the General position transformation matrix to the manipulator transformation matrix. In some cases this can be a simplified matrix 2.Look at both matrices for: 1.Elements witch contain only one joint variable 2.Paris of elements which can be used to produce an equation in only one joint variable. (look for divisions that will result in atan2 functions. 3.Elements or combinations of elements that can be simplified with trig identities 3.Having selected an element, equate it to the corresponding element in the position matrix. Solve for desired variable

Inverse Kinematics Heuristic Cont. 4.Repeat 3 until all elements are identifies. 5.If solutions suffer from inaccuracies or redundant results look for alternatives 6.If all joints cannot be found pre-multiply both sides by the inverse of the first links transformation matrix that is not solved. Repeat starting at step 1. 7.Continue repeating steps 2-6 until all solutions are found. 8.If no solution can be found for all the variables. The position matrix may not be in the manipulators workspace.

Definitions Redundancy: When a robot can reach a position with more than one position of its linkage this is considered a redundancy. Certain mathmatical formulas in the solution will point to this –Square root or Cosine function –Examples Degeneracy: If an infinite number of solution achieve the position the manipulator has a degeneracy. This is also a position where control over one or more degrees of freedom are lost. RR equal lengths.

Introduction to Control Open Loop Closed Loop PID Feed Forward (Dynamic Model)

Vocabulary Plant: The machine, process or device being controlled. Processes: Any operation to be controlled. Set-Point: A desired value for the position, velocity or other control variable. Error: Difference between the current value and set-point of a variable. Feedback Control: Maintains a relationship between an output and a set-point by comparing these and using the difference as a means of control.

Open Loop Relies on plant to perform in a known way. –I.E. Move twice as far if you turn it on twice as long. No feed back term: The output has on effect on the controlling action. Don’t measure to see if you got their. If system is stable it will always be stable. If you can use you should. Examples –Washing Machine –Blender –Throwing a ball –Driving to a hardtop.

Closed Loop Control Output has a direct effect on the input. Uses Feedback to reduce error. Can reject disturbances. Can be used to reject disturbances, or changes in the plant. Can get by with cheaper components. Stability can be an issue Examples –Oven –Second Ball thrown –Threading a needle –Motor control –

Closed Loop Control Bang-Bang: Applies full command until an event occurs. –A Light Switch. –Garage Door Opener –Thermostat Proportional: Applies a command proportional to an error or combinations of variables. –A Dimmer Switch –Motor Controller –Elevators Command Actual Position Desired Position

PID Control Proportional Command Is Proportional to the difference between the desired set-point and the current set-point, i.e. the error e(t). Integral Output is changed at a rate proportional to the actuating error signal e(t) Derivative Control action is proportional to the rate of change of the error signal e(t)

PID Control Proportional –Is an amplifier with an adjustable gain Integral –Removes Steady state error. –Can destabilize a system Derivative –Has an anticipatory action –Amplifies noise in the signal –May cause saturation in the actuator. –Why can it not be used alone. em - +

Feed Forward Predict what command should be used in advance Requires model of the system –Can Cause instability if model is not good –Calculate in Real-Time –Lookup Table (Gain Scheduling) Reduces following error Allows lower system Gain –Better Tracking –Better Stability Command Actual Position Desired Position Feedback Feed forward Plant em - + Model + + CommandOutput

Torque Loop Simplest Control of Motor –Torque Amplifier –Control Force out of motor Independent of Speed Independent of Position ECurrent - + Amp motor Command Load

Control of Robots

Level of Control Cartesian –Linear –Model Based Joint Servo

Types of Control Open Loop –No feedback Steppers –No disturbance rejection Closed Loop –Linear –Non-linear Adaptive Cartesian de-coupling

Vocabulary Linear: Based on linear differential equations. Open Loop: Closed Loop:

Open Loop Joint and Robot Control Trajectory Generator Control System Needs Perfect model No disturbance rejection Hope it follows Path Stepper motors would be an example

Closed Loop Joint Control Open Loop Robot Trajectory Generator Control System Trajectory is a function of only the desired trajectories Common in some industries If we have good model and no disturbances we get good following

Closed Loop Cartesian Control Coordinate Conversion and Control System Kinematics - + Closes loop along the path.

Variant of Closed Loop Cartesian Control Control System Kinematics - + Closes loop along the path. Errors must be small for Jacobian to work

Comparison Between Joint and Cartesian Control Computationally complex Rejects Disturbances at joints and along path Can follow Straight Lines and Complex trajectories Inverse and Direct Kinematics need to be performed. Errors in model may limit gains Simple Rejects Disturbances only at Joint Follows Joint Profile, will have not necessarily limit path error. CartesianJoint