Experiment 3: Inverse Kinematics (IK) for the Cobot-C1

Determine the joint angles required for the end-effector to reach a desired position and orientation.

Objectives

Build proficiency in controlling the Cobot-C1 by commanding end-effector target positions
Link theory and hands-on application by applying mathematical IK concepts to a real robot
Test and compare various IK solutions for selected end-effector poses
Bridge the gap between task space goals and joint space commands

Key Concepts

Inverse Kinematics: Reverse of forward kinematics - finding joint angles from desired pose
Task Space to Joint Space: Converting Cartesian goals into joint commands
Analytical vs. Numerical Methods: Different approaches to solving IK equations
Multiple Solutions: Handling configurations like elbow-up vs. elbow-down
Workspace Limitations: Understanding unreachable poses

Theory

What is Inverse Kinematics?

Inverse kinematics (IK) is a fundamental problem in robotics involving kinematic chains. It is the mathematical procedure used to determine the joint variables (angles or displacements) required to achieve the desired position and orientation of the robot's end-effector relative to its base.

Unlike forward kinematics, which deterministically computes the pose given joint values, inverse kinematics works in reverse: it calculates the necessary joint configurations to realize a known end-effector pose. This is crucial because actions are performed by the end-effector, but control inputs are applied at the joints.

How Does Inverse Kinematics Work?

Input: Desired end-effector position and orientation (pose in 3D space)
Output: One or more sets of joint variables that position the end-effector at the target
Process: Solve nonlinear equations derived from forward kinematics
Multiple or No Solutions: Different joint configurations may produce the same pose; some poses may be unreachable
Solution Methods: Analytical (geometric/algebraic) or numerical (Newton-Raphson, Jacobian-based, optimization)

Inverse Kinematics Equations

Note: The following equations use a simplified 4-DOF model for clarity. The concepts extend to the full 6-DOF Cobot-C1.

Given

Desired end-effector pose: (x, y, z, φ)

Goal

Find joint angles: θ₁, θ₂, θ₃, θ₄

Solution Steps

Step 1 - Base Joint Angle

θ₁ = arctan2(y, x)

Step 2 - Effective Reach and Vertical Offset

r = √(x² + y²) z' = z - L₁

Step 3 - Distance from Shoulder to Wrist

D = √(r² + (z')²)

Step 4 - Elbow Joint (Law of Cosines)

cos(θ₃) = (D² - L₂² - (L₃ + L₄)²) / (2L₂(L₃ + L₄)) θ₃ = arccos((D² - L₂² - (L₃ + L₄)²) / (2L₂(L₃ + L₄)))

Step 5 - Shoulder Joint (Planar Triangle Geometry)

α = arctan2(z', r) β = arccos((L₂² + D² - (L₃ + L₄)²) / (2L₂D)) θ₂ = α - β

Step 6 - Wrist Joint (End-Effector Orientation)

θ₄ = φ - θ₂ - θ₃

Solution Steps Summary

Step	Description
1	Compute θ₁: arctan2(y, x)
2	Find planar distance r: √(x² + y²)
3	Compute z': z - L₁
4	Use law of cosines for θ₃ (elbow)
5	Use geometry and law of sines/cosines for θ₂ (shoulder)
6	Solve orientation constraint for θ₄ (wrist)

Prerequisites

Connection of the Cobot-C1 robot to the workstation and initialization of software stack
Calibration of joint encoders and configuration of end-effector position readout
Completion of environment and safety checks for the workspace
Availability of code or GUI tools for inputting and recording positions and angles
Review of basic forward kinematics concepts and familiarity with coordinate convention

Procedure

Step 1: Launch the EE (XYZ) Control GUI

cd dev/MRL/cobot-C1/src/cobo_control/src/ python3 indendcontrol.py

Step 2: Safety Preparation

Wait for automatic movement to safe startup position
Verify GUI status shows "Ready for safe EE control"
Ensure workspace is clear

Step 3: Move the Cobot Using End-Effector Controls

Use GUI and keyboard (WASDQE keys) to move end-effector in Cartesian (X, Y, Z) space
Joints 4, 5, and 6 are locked for safety
Move incrementally and observe EE position values in GUI

Step 4: Record End-Effector Position and Joint Angles

Record EE Position from GUI (X, Y, Z in mm, phi and psi/orientation)
Press J key to print all six joint angles (degrees/radians)
Write down both EE pose and joint angles

Step 5: Use IK Equations to Validate Joint Angles

Apply the IK equations to calculate expected joint angles from the recorded end-effector position.

Step 6: Compare Results

Compare IK-derived angles with actual recorded angles and note any discrepancies.

Step 7: Repeat

Repeat steps 2-6 for different end-effector poses throughout the workspace.

Observations Table

Trial	Recorded Joint Angles	Recorded EE Location	Calculated Joint Angles (IK)	Notes
1
2
3
4
5
6

Sources of Error and Precautions

Common Sources of Error

Measurement Inaccuracies: Mistakes when recording EE position or joint angles; lag or drift may affect readings
Numerical Approximation: Rounding errors during IK calculations
Model Simplifications: Mathematical models may ignore mechanical flex or sensor offsets
Unreachable Poses: Commanding EE positions outside robot's workspace results in no solution
Multiple IK Solutions: Some EE positions correspond to multiple valid joint configurations

Precautions

Verify input values: Double-check all EE positions and joint angles, note correct units
Move safely: Perform small, incremental EE moves and visually confirm robot configuration
Stay within workspace: Ensure all EE poses are within specified workspace
Repeat measurements: For each pose, repeat measurements for consistency
Interpret IK solutions carefully: For multiple solutions, select the one matching the real robot arrangement

Learning Outcomes

Appreciate the principles of inverse kinematics for serial manipulators
Recognize methods for determining joint values required to achieve specific end-effector poses
Identify challenges in solving inverse kinematics, such as multiple solutions and workspace limitations
Understand the interplay between analytical solutions and numerical approaches
Observe typical discrepancies between ideal models and hardware outcomes