Unit 5: Inverse Kinematics - Subjective Questions

Inverse kinematics (IK) is the process of determining the joint variables required to place the end-effector at a specified position and orientation.

For a two-axis planar articulated robot:

• Forward kinematics calculates the end-effector coordinates from known joint angles:
  
• Inverse kinematics calculates the joint angles from a desired end-effector position:
  

Forward kinematics generally gives one end-effector position for a given set of joint angles. Inverse kinematics may produce multiple solutions, no solution, or infinitely many solutions depending on the target position and robot configuration.

A two-axis planar articulated robot consists of two rotary joints connected by two rigid links. The motion of both links is restricted to a single plane.

• The first joint rotates the first link by angle relative to the reference axis.
• The second joint rotates the second link by relative angle with respect to the first link.
• The link lengths are denoted by and .
• The end-effector is attached to the distal end of the second link.

The arm reaches a point by combining the rotations of both joints. The Orangewood Robotic Arm demonstrates this principle using servo motors, a controller, and a gripper or end-effector. The controller converts calculated joint angles into servo commands so that the end-effector reaches the desired planar location.

Consider a planar robot with two revolute joints. The first link makes an angle with the positive -axis. The second link has an absolute orientation of .

The coordinates contributed by the first link are:

The coordinates contributed by the second link are:

Therefore, the end-effector position is:

These equations form the basis for solving the inverse kinematics problem.

The geometric approach uses the triangle formed by the two links and the line connecting the base to the target point .

Let the distance from the robot base to the target be:

The angle made by the target vector with the positive -axis is:

Using the law of cosines, the elbow angle is determined from:

Thus:

The first joint angle is found by subtracting the internal triangle angle from :

The positive and negative signs correspond to the two possible configurations: elbow-up and elbow-down.

For a desired end-effector position , the distance between the base and the target is:

The links , , and form a triangle. Applying the law of cosines gives:

Rearranging:

Since :

The two possible values of the joint angle are:

The two solutions represent the elbow-up and elbow-down configurations.

The direction of the line joining the base to the target point is:

The second angle required in the triangle is determined from the vector formed by the two-link chain. Its value is:

The first joint must rotate through the target direction minus this triangle angle. Hence:

Substituting and gives:

The selected value of determines whether the resulting configuration is elbow-up or elbow-down. The function is preferred because it correctly handles the quadrant of the target point.

For many reachable target points, a two-link planar robot can reach the same point using two different postures.

• In the elbow-up configuration, the second joint bends in one direction and the intermediate joint lies on one side of the line joining the base and target.
• In the elbow-down configuration, the second joint bends in the opposite direction and the intermediate joint lies on the other side of that line.

Mathematically, the two configurations arise from:

and

where:

Both solutions produce the same end-effector position, but they may differ in collision avoidance, joint limits, reachability of nearby points, and motion smoothness.

Let the target point be and its distance from the base be:

A target is reachable if it lies within the annular workspace defined by the link lengths:

Equivalently, the inverse kinematics calculation must satisfy:

If the value is outside this range, no real value of exists and the target cannot be reached. The outer boundary occurs when the links are fully extended, while the inner boundary occurs when the links fold back over one another.

A singular configuration occurs when the robot loses the ability to generate motion in one or more Cartesian directions through small joint movements.

For a two-link planar robot, singularities commonly occur when:

• The two links are completely stretched, so .
• The two links are folded back on one another, so or .

The Jacobian matrix is:

Its determinant is:

Therefore, the robot is singular when . Near a singularity, small Cartesian movements may require very large joint movements, causing poor control accuracy and high actuator demands.

The Denavit-Hartenberg convention represents a serial robot using homogeneous transformation matrices between successive coordinate frames. Each link is described using four parameters:

• Link length : distance along the -axis between the -axes.
• Link twist : angle between successive -axes measured about the -axis.
• Link offset : distance along the -axis.
• Joint angle : rotation about the -axis.

The standard DH transformation is:

and its matrix form is:

One common standard DH representation for a two-link planar robot is:

The interpretation is:

• Both joint axes are parallel and perpendicular to the plane of motion.
• Therefore, the link twists are .
• The robot has no fixed displacement along the joint axes, so .
• The link lengths are and .
• Since both joints are revolute, and are the variable joint parameters.

Depending on the frame assignment and DH convention used, equivalent tables may have slightly different indexing, but they must produce the same physical transformation.

For the two-link planar robot, the DH matrices are:

The complete transformation is:

Multiplying the matrices:

The last column gives the end-effector position, while the upper-left block describes its planar orientation.

The geometric and DH approaches solve the same robot problem but differ in method and applicability.

For the Orangewood Robotic Arm, the geometric approach is useful for understanding the two-link motion, while the DH method provides a formal modeling procedure that can be extended to more joints and spatial configurations.

Implementation involves converting a desired Cartesian position into servo joint commands.

• Measure or obtain the arm link lengths and .
• Specify the target coordinates in the robot base frame.
• Calculate:
  
• Check whether lies within .
• Compute the selected elbow configuration:
  
• Compute the first joint angle:
  
• Convert angles from radians to degrees if required by the servo controller.
• Apply servo zero offsets, direction corrections, and calibration factors.
• Limit the commands to the allowable servo ranges before sending them to the Orangewood Robotic Arm.

The calculated position should be verified through forward kinematics after the servo commands are applied.

The ordinary inverse tangent function cannot uniquely determine the quadrant of the target point because the ratio may be the same in multiple quadrants.

The function uses the signs of both and to determine the correct angle over the complete range:

Its advantages include:

• Correct quadrant determination.
• Proper handling of points where .
• Improved numerical robustness.
• More accurate calculation of the target direction .

In the inverse kinematic equation:

using helps ensure that the robot selects the correct physical posture for the target location.

Given:

First calculate:

Then:

Therefore:

The two approximate values are:

The target direction is:

For the positive solution:

For the negative solution:

Thus, the approximate configurations are:

• Configuration 1:
• Configuration 2:

Mathematical inverse kinematics may produce solutions that are physically impossible for the actual robot. Joint limits and actuator constraints must therefore be checked after calculating the angles.

Important constraints include:

• Minimum and maximum rotation angles of each servo.
• Mechanical stops in the robot links.
• Maximum servo torque and payload capacity.
• Servo resolution and dead-band.
• Required speed and acceleration.
• Collision restrictions between links, the base, and the workpiece.

A solution is acceptable only if:

for every joint. If one solution violates a joint limit, the other elbow configuration may be selected. If both solutions violate the constraints, the target should be rejected or modified. Practical implementations should also include angle offsets caused by servo mounting and calibration.

Validation checks whether the calculated joint angles actually produce the desired end-effector position.

The procedure is:

• Specify the desired target .
• Calculate the joint angles using inverse kinematics.
• Substitute these angles into the forward kinematic equations:
  
  
• Compare with .
• Calculate the position error:
  

If is within the specified tolerance, the solution is considered valid. In a practical Orangewood Robotic Arm experiment, the calculated angles should also be sent to the arm and the actual end-effector position should be observed or measured. Differences may occur because of backlash, servo offset, link flexibility, or inaccurate link measurements.

Inverse kinematics assumes that the robot geometry and joint angles are known accurately. In practice, errors in these quantities cause the actual end-effector position to differ from the calculated position.

Sources of error include:

• Incorrect measurements of and .
• Servo zero-position offset.
• Difference between commanded and actual joint angles.
• Gear backlash and mechanical play.
• Link bending under load.
• Inaccurate coordinate-frame alignment.
• Quantization of servo commands.

A small angular error can create a larger Cartesian error, especially when the arm is extended or near a singular configuration. Calibration should therefore establish the true zero angle and the relationship between command values and physical angles. The measured link dimensions should be used in the kinematic model, and the final position should be corrected using experimental testing.

A homogeneous transformation matrix combines rotation and translation in a single matrix. It describes the position and orientation of one coordinate frame relative to another.

A general transformation is written as:

where is a rotation matrix and is a position vector.

For a serial robot, individual link transformations are multiplied:

For the two-axis planar robot, gives the pose of the end-effector relative to the base. Its position components provide and , while its rotational portion gives the end-effector orientation. Homogeneous matrices make it possible to model several links systematically and are especially valuable when the robot is extended from a planar two-link system to a spatial manipulator.