U21BMP37�ROBOTICS IN MEDICINE

Dr. N. Rajasingam

UNIT - II - KINEMATICS

Inverse Kinematics - General properties of solutions tool configuration - Four axis - Workspace analysis - Trajectory planning work envelope - Examples - Workspace fixtures - Pick and place operations - Continuous path motion - Interpolated motion and straight-line motion.

CO2: Elucidate the kinematics, motion planning and control involved in design of robotic systems (Understand)

Direct Kinematics

• The key to the solution of the direct kinematics problem was the Denavit-Hartenberg (D-H) algorithm, a systematic procedure for assigning link coordinates to a robotic manipulator.
• Successive transformations between adjacent coordinate frames, starting at the tool tip and working back to the base of the robot, then led to the arm matrix.

• The arm matrix represents the configuration of the tool [position p and orientation R] in the base frame as a function of the joint variables q.

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• The solution to the direct kinematics problem can be expressed as

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• R represents the rotation
• p represents the translation of the tool frame relative to the base frame

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Inverse Kinematics

• Direct kinematics: procedure for determining the position and orientation of the tool of a robotic manipulator by given vector of joint variables.
• Inverse kinematics: determining the joint variables by given desired position and orientation for the tool.

Relationship between the Kinematics

• In joint space, the vector of joint variables q is restricted to the subset of Rⁿ.
• The tool-configuration parameters {R, p} can be associated with a subset of R⁶.
• Tool-configuration space is six-dimensional because arbitrary configurations of the tool can be specified by using three position coordinates (p₁, p₂, p₃) together with three orientation coordinates (yaw, pitch, roll).

• Solving the direct kinematics problem is equivalent to finding the mapping from joint space to tool-configuration space, while solving the inverse kinematics problem is equivalent to finding an inverse mapping from tool-configuration space back to joint space.
• The inverse kinematics problem is important because manipulation tasks are naturally formulated in terms of the desired tool position and orientation.

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• Example: when external sensors such as overhead cameras are used to plan robot motion, where the information provided by the camera is not in terms of joint variables; it specifies the positions and orientations of the objects that are to be manipulated.
• A compact representation of tool position and orientation called the tool- configuration vector is introduced where the inverse kinematics problem is more difficult than the direct kinematics problem because a systematic closed-form solution applicable to robots in general is not available.

General Properties of Solutions

• An explicit solution [find values for the joint variables q which satisfy the arm equation] to the inverse kinematics problem depend upon the robot or the class of robots being investigated.
• Unique solutions are rare.
• Multiple solutions exist.

Multiple solutions: nonredundant robot

Solvability

General Properties of Solutions

• 12 equations
• n unknowns [ n=6 for six-axis robot]
• Out of 9 equations in rotation matrix, 3 are independent [normal, sliding, approach vectors]
• 3 equations in position matrix are independent.
• Total 6 independent equations in Transformation matrix.

Existence of solutions

• A manipulator is solvable if all the sets of joint variable can be found corresponding to a given end-effector location.
• Conditions:
• Tool point within the workspace
• Robot has more than or equal to three degrees of freedom to orient the tool [n>=6]
• None of the joint limitations are violated

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Existence of solutions

• Kinds of solution:
• Closed form solutions [algebraic or geometric approach]
• Numerical solutions [iterative search]
• Closed form solutions: Three adjacent joint axes intersecting or parallel to each other.

Uniqueness of Solutions

• Infinitely many solutions to the inverse kinematics problem typically exist for robots with more than six axes as kinematically redundant robots, because they have more degrees of freedom.
• The extra degrees of freedom add flexibility to the manipulator.

Reaching around an obstacle with a redundant robot

• Example: a redundant robot might be commanded to reach around an obstacle and manipulate an otherwise inaccessible object.
• Here some of the degrees of freedom can be used to avoid the obstacle while the remaining degrees of freedom are used to configure the tool.

Tool-configuration

• tool-configuration vector

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Link coordinates of a four-axis SCARA

Elbow Joint

Base Joint

Vertical Extension Joint

Tool Roll Joint

Algorithm

Workspace analysis

• Detailed analysis of the work envelope based on the positions reachable by the tool tip.
• In using the tool tip as a reference point, the effects of both the major axes used to position the wrist and the minor axes used to orient the tool has to be included.

Work Envelope of a Four-axis SCARA

Tool Vector

Joint Limits

Maximum and minimum horizontal reach of a SCARA robot

Horizontal cross section of restricted SCARA work envelope

Workspace fixtures

• Workspace fixtures are accessories and add-on of a robot.
• These fixtures form a part of a robot to perform a particular task.
• Work space fixtures used in the work space of the robot are:
• Part feeders 
• Transport devices 
• Part-holding devices

Concentric layout of multiple part feeders 

Gravity-fed part feeder

Linear Transport devices (Conveyor belts)

Rotary transport device (carousel)

Part-holding devices

• Devices which are used to hold a sub-part in exact position and orientation.
• It is mainly used in assembly line task.
• Most robots are one-handed and have only one tool in operation at any given time.
• A work-holding fixture is considered as a fixed tool.

Pick-and-Place operation

• Most fundamental robotic manipulation task.
• Example: automated loading and unloading of machines, alter the distribution of parts within the workspace.
• Simple point-to-point motion control can usually be used to execute pick-and-place operations, with the minimal pick-and-place trajectory consisting of four discrete points.

Pick-and-Place Trajectory

Speed Variation

Place distance constraint

Continuous path motion

• If the speed of each joint of the robot can be controlled independently, then the robotic manipulator is capable of continuous-path motion control as opposed to the more primitive point-to-point motion control.
• Required in those applications where the tool must follow a specific path between points in the workspace.
• Typically, the speed with which the tool moves along that path must also be regulated.
• Examples: paint spraying, are welding and the application of glue or sealant.

Interpolated motion: Piecewise-linear interpolation between knot points

Interpolated motion: Piecewise-linear interpolation with a parabolic blend

Straight-line motion

Taylor's algorithm for approximating straight-line motion with a point-to-point robot