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**Review: Homogeneous Transformations**

Homogeneous vector Homogeneous transformation matrix

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**Review: Aim of Direct Kinematics**

Compute the position and orientation of the end effector as a function of the joint variables

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**Review: Direct Kinematics**

The direct kinematics function is expressed by the homogeneous transformation matrix

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Review: Open Chain Computation of direct kinematics function is recursive and systematic

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**Review: Denavit-Hartenberg Convention**

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**Review : D-H Convention**

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**Class Problem: Spherical Wrist**

1. Fill in the table of D-H parameters for the spherical wrist. 2. write the three D-H transformation matrices (one for each joint) for the spherical wrist 3. Find the overall transformation matrix which relates the final coordinates (x6y6z6) to the “base” coordinates (x3y3z3) for the spherical wrist

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**Review : D-H Convention**

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**Joint Space and Operational Space**

Description of end-effector task position: coordinates (easy) orientation: (n s a) (difficult) w.r.t base frame Function of time Operational space Joint space Prismatic: d Revolute: theta Independent variables

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**Joint Space and Operational Space**

Direct kinematics equation Three-link planar arm (Pp ) M的最大值是？平面运动只需要3个自由度

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**Joint Space and Operational Space**

Generally not easy to express

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**Joint Space and Operational Space**

Workspace reachable workspace dexterous workspace Factors determining workspace Manipulator geometry Mechanical joint limits Mathematical description of workspace Workspace is finite, closed, connected

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Workspace Example

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**Performance Indexes of Manipulator**

Accuracy of manipulator Deviation between the reached position and the position computed via direct kinematics. repeatability of manipulator A measure of the ability to return to a previously reached position.

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**Kinematic Redundancy Definition**

A manipulator is termed kinematically redundant when it has a number of degrees of mobility which is greater than the number of variables that are necessary to describe a given task.

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**Kinematic Redundancy Intrinsic redundancy m<n functional redundancy**

relative to the task Why to intentionally utilize redundancy?

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**Kinematic Calibration**

Kinematic calibration techniques are devoted to finding accurate estimates of D-H parameters from a series of measurements on the manipulator’s end-effector location. Direct measurement of D-H is not allowed.

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Inverse Kinematics

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Inverse Kinematics we know the desired “world” or “base” coordinates for the end-effector or tool we need to compute the set of joint coordinates that will give us this desired position (and orientation in the 6-link case). the inverse kinematics problem is much more difficult than the forward problem!

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Inverse Kinematics there is no general purpose technique that will guarantee a closed-form solution to the inverse problem! Multiple solutions may exist Infinite solutions may exist, e.g., in the case of redundancy There might be no admissible solutions (condition: x in (dexterous) workspace)

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Inverse Kinematics most solution techniques (particularly the one shown below) rely a great deal on geometric or algebraic insight and a few common “tricks” to generate a closed-form solution Numerical solution techniques may be applied to all problems, but in general do not allow computation of all admissible solutions

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Three-link Planar Arm x is known, compute q

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Three-link Planar Arm W can be expressed both as a function of end-effector p&o, and as a function of a reduced number of joint variables

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Class problem Two-link planar arm one-link planar arm

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Three-link Planar Arm Algebraic approach

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Three-link Planar Arm no admissible solution If c2 is out of this range Elbow up and elbow down

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Three-link Planar Arm 对c12和s12用三角公式展开，则得二元方程组，变量为s1和c1。

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**Three-link Planar Arm Geometric approach l**

Feasible condition: a1+a2>l and |a1-a2|<l

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Class Problem what are the forward and inverse kinematics equations for the two-link planar robot shown on the right? Y0 2nd Joint: Prismatic 90 deg Attention: m= ? 1st Joint: Revolute X0

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