Page 148 - Toucpad robotics C11

The manipulator shown in the diagram appears to have multiple degrees of freedom at its waist, shoulder (between
lower and upper arm), elbow (between upper arm and wrist), and wrist, allowing for versatile movements similar to
a human arm.
Mathematical Concepts for Analysing Manipulator Arms

Understanding and controlling a robotic arm requires a precise mathematical description of its structure and motion.
The two core mathematical concepts are kinematics and dynamics. We have touched upon them briefly in the
“Physics & Mathematics” section, but here we will focus specifically on their application to manipulators.
Kinematics of Manipulators

Definition: Kinematics deals with the geometry of motion without considering the forces that cause the motion.
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For manipulators, it’s about the relationship between the joint angles and the position/orientation of the end-effector.
Forward Kinematics

u Concept: Given the known lengths of the robot’s links and the angles of all its joints, forward kinematics calculates
the exact position (x, y, z coordinates) and orientation (e.g., pitch, roll, yaw angles) of the robot’s end-effector
in space.

u Mathematical Tool: This involves using transformation matrices (as discussed in the ‘Matrix Operations’ section)
to represent the position and orientation of each joint’s coordinate frame relative to the previous one. By multiplying
these matrices sequentially from the base to the end-effector, the final position and orientation of the end-effector
relative to the robot’s base can be determined.
u Practical Use: This is used by the robot’s controller to know exactly where its tool is at any given moment, based on
the sensor readings of its joint angles.
u Example: If a robotic arm needs to know if its gripper is currently above a specific part on a conveyor belt, it uses
forward kinematics to calculate the gripper’s real-time position from its joint encoder readings.
Inverse Kinematics
Concept: This is the more complex and common problem in robotics. Given a desired position and orientation for the
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robot’s end-effector (where you want the robot’s tool to be), inverse kinematics calculates the specific angles that
each of the robot’s joints must take to achieve that desired position and orientation.
Mathematical Challenge: For many robotic arms, solving inverse kinematics involves complex trigonometric
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equations or iterative numerical methods, as there might be multiple possible joint configurations to reach the same
end-effector position (e.g., an elbow-up or elbow-down configuration for a human arm).
Practical Use: This is what makes a robot usable. When a programmer tells a robot to “pick up object A,” they specify
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the location of object A in the workspace, not the individual joint angles. The robot’s control system then uses inverse
kinematics to figure out the required joint angles to reach that object.
Example: When programming a welding robot to follow a specific seam, the programmer defines the path of the
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weld in space. The robot’s controller continuously solves the inverse kinematics problem to determine the joint angles
required at each point along the seam to guide the welding torch precisely.

Dynamics of Manipulators
Definition: Dynamics deals with the relationship between the forces and torques acting on a manipulator and the
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resulting motion (acceleration, velocity). It considers the mass, inertia, and friction of the robot’s links and the loads
it carries.
Practical Use:
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Motor Sizing: Dynamics helps engineers determine the required power and torque ratings for the motors at each
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joint to move the manipulator and its payload efficiently. If the motors are too weak, the arm won’t be able to
move or lift the required load. If they are too strong, they add unnecessary weight and cost.

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