Unit 6: I’m organizing the notes around the planning method, Cartesian straight-line generation, the two-link planar kinematics, and the Orangewood arm implementation. The equations will distinguish Cartesian path generation from joint-space execution, since that distinction is central to obtaining a genuinely straight end-effector path.# Unit 6: Trajectory Planning - Subjective Questions

Trajectory planning determines the position, velocity, and acceleration of the robot joints or end-effector as functions of time.

• A path specifies only the geometric route between the initial and final points.
• A trajectory adds timing information to the path.
• Path planning answers: Where should the robot move?
• Trajectory planning answers: Where should it be at each instant, and how fast should it move?

For a straight-line Cartesian path from to , the path can be written as

A timing law converts this path into a trajectory. The corresponding joint angles and are then obtained using inverse kinematics.

The two-axis planar model consists of two revolute joints and two rigid links moving in a common plane.

• The first joint rotates the first link by .
• The second joint rotates the second link relative to the first by .
• The link lengths are represented by and .
• The end-effector position is represented by .

The forward-kinematic equations are

This simplified model is used to compute the joint commands required for straight-line end-effector motion. During implementation, the physical arm's joint offsets, motion plane, joint limits, and link dimensions must also be considered.

Let the first and second link lengths be and , and let the joint variables be and .

The endpoint of the first link is

The orientation of the second link with respect to the base frame is . Its Cartesian components are

Adding the components gives the end-effector position:

These equations map joint-space coordinates to Cartesian coordinates . They are used to verify whether the executed trajectory follows the required straight line.

For a two-link planar robot,

Squaring and adding these equations gives

Therefore,

Let

Then

The first joint angle is

The positive and negative choices of produce the elbow-up and elbow-down configurations. A valid solution must satisfy and the physical joint limits of the Orangewood Robotic Arm.

Let the initial and final end-effector positions be

A straight-line path is generated by linear interpolation:

where is a path parameter.

• At , the end-effector is at .
• At , it is at .
• Intermediate values of produce points on the line segment.

For equal path intervals,

Each point is converted into joint angles using inverse kinematics. A time-scaling function may replace uniform values when smooth acceleration and deceleration are required.

Joint interpolation defines

for each joint . However, the relationship between joint angles and Cartesian position contains nonlinear trigonometric terms:

Consequently, equal changes in joint angles do not correspond to equal Cartesian displacements. The end-effector usually follows a curved path even though each joint angle varies linearly.

To obtain straight-line motion:

• Interpolate points in Cartesian space.
• Apply inverse kinematics to every interpolated point.
• Select a continuous inverse-kinematic branch.
• Command the resulting joint sequence with suitable timing.

Joint-space planning:

• Interpolates and directly.
• Is computationally simple.
• Makes joint limits and joint-rate constraints easier to apply.
• Does not generally guarantee a straight end-effector path.

Cartesian-space planning:

• Interpolates the end-effector coordinates .
• Can guarantee geometric straightness when sufficiently sampled and accurately executed.
• Requires inverse kinematics at every trajectory point.
• Must handle unreachable points, multiple solutions, singularities, and joint limits.

For the stated laboratory task, Cartesian-space interpolation is appropriate because the required motion is explicitly a straight line. The resulting Cartesian points are transformed into joint commands before being sent to the arm.

A practical implementation follows these steps:

1. Measure or configure , , joint zero offsets, joint limits, and command units.
2. Define the initial point and final point in the robot base frame.
3. Check whether both points and the line segment are inside the reachable workspace.
4. Select the motion duration , sampling interval , and time-scaling law .
5. Generate Cartesian samples using

6. Solve inverse kinematics for every sample.
7. Select a continuous elbow-up or elbow-down branch and enforce joint limits.
8. Convert mathematical joint angles into calibrated actuator commands.
9. Send time-stamped commands to the robot controller at the required update rate.
10. Monitor execution and stop the arm if communication, limit, or tracking errors occur.
11. Use forward kinematics or measured feedback to evaluate the actual end-effector path.

The trajectory should be validated at low speed before operation at the intended velocity.

Time scaling maps time to a normalized path parameter . It allows the robot to follow a fixed geometric path with a specified velocity profile.

Assume

with boundary conditions

From the first two conditions at ,

Using the final conditions gives

Thus,

Its velocity is

The Cartesian trajectory is then

This timing law starts and ends with zero velocity, reducing abrupt motion compared with constant-velocity commands.

Let the normalized time be

A quintic polynomial is selected to satisfy six boundary conditions:

Solving for the polynomial coefficients gives

where . Its derivatives are

The Cartesian position is calculated from

Quintic scaling provides smoother endpoint behavior than cubic scaling because both velocity and acceleration become zero at the start and end.

A digital robot controller executes a finite sequence of trajectory samples rather than a mathematically continuous curve.

If the duration is and the sampling interval is , the approximate number of intervals is

The Cartesian distance between neighboring samples for uniform interpolation is approximately

• A smaller produces more waypoints and a better approximation of continuous straight-line motion.
• A very large can cause visible segmenting, vibration, and larger tracking errors.
• A very small increases computation and communication load.
• The update rate must be supported by the Orangewood controller and communication interface.

Sampling must therefore balance path accuracy, actuator smoothness, computation time, and reliable command transmission.

For a desired point , define its radial distance from the base as

For an ideal two-link planar robot, the point is reachable when

Equivalently, the inverse-kinematic term

must satisfy

Checking only the initial and final points is insufficient. Every interpolated point on the line segment should be tested because physical joint limits, collision constraints, or restricted operating regions may make an intermediate point invalid.

A practical program should reject the trajectory before motion if any point is unreachable, outside a joint limit, or unsafe.

For many reachable Cartesian points, a two-link planar robot has two possible configurations. They arise from

• One sign produces the elbow-up solution.
• The other sign produces the elbow-down solution.

Both solutions place the end-effector at the same Cartesian point but use different joint angles.

A branch should be selected by considering:

• The robot's initial configuration.
• Joint limits and mechanical clearances.
• Collision avoidance.
• Distance from singular configurations.
• The magnitude of required joint motion.

During a trajectory, the selected solution must remain continuous. One common method is to calculate all valid solutions at each waypoint and choose the solution closest to the previous joint vector:

Uncontrolled branch switching can create large and unsafe jumps in commanded joint angles.

The forward kinematics are

Differentiating with respect to the joint angles gives

where

The Jacobian is used to:

• Relate joint velocities to Cartesian velocities.
• Calculate joint rates required for a desired end-effector velocity.
• Detect singular configurations.
• Check whether joint-speed limits will be exceeded.

When the Jacobian is nonsingular,

may be used for differential trajectory control.

A kinematic singularity occurs when the Jacobian loses rank. For the two-link planar robot,

Therefore, singularities occur when

or

These configurations correspond to the links being fully extended or folded back along the same line.

Near a singularity:

• A small Cartesian velocity may require very large joint velocities.
• Numerical inverse calculations become poorly conditioned.
• Tracking errors and vibrations may increase.
• The inverse-kinematic solution branches may meet or become difficult to distinguish.

The planner should detect near-singular configurations, reduce Cartesian speed, choose another feasible configuration, modify the path, or use a regularized inverse such as a damped pseudoinverse.

After calculating the joint sequence , finite differences can estimate joint velocity and acceleration:

The trajectory is acceptable only if

and

for both joints at all samples.

If a limit is violated, the planner may:

• Increase the total motion time .
• Apply a smoother time-scaling law.
• Reduce Cartesian speed near singularities.
• Increase waypoint density for more reliable evaluation.
• Modify the path if the required joint motion remains infeasible.

These checks protect the actuators and improve trajectory tracking.

The angles produced by inverse kinematics are mathematical angles in the robot model. They may not directly match the actuator command convention.

The implementation should determine:

• The physical zero position of each joint.
• The positive direction of rotation.
• The conversion between radians, degrees, encoder counts, or controller units.
• Gear ratios, if they are not handled by the controller.
• Mechanical and software joint limits.
• The transformation between the laboratory frame and robot base frame.

A typical command conversion is

where represents direction and unit scaling, and is the calibrated zero offset.

Calibration can be verified by commanding low-speed test angles, measuring the resulting end-effector position, and comparing it with forward-kinematic predictions. Incorrect offsets or signs can produce a shifted, mirrored, or unsafe trajectory.

Let the desired line start at and end at . Define

For a measured end-effector point , its perpendicular distance from the desired line is

Useful performance measures include:

• Maximum cross-track error:

• Root-mean-square error:

• Final position error .
• Joint tracking error between commanded and measured angles.
• Total execution-time error.

Measured joint feedback may be converted to Cartesian coordinates using forward kinematics. For stronger validation, an external camera or position-measurement system can independently record the end-effector path.

Important error sources include:

• Incorrect measurements of and .
• Joint zero-offset and direction errors.
• Encoder quantization and actuator command resolution.
• Gear backlash, compliance, and friction.
• Controller latency and nonuniform communication timing.
• Low command update rate or widely spaced waypoints.
• Inverse-kinematic numerical errors.
• Servo tracking lag during rapid acceleration.
• Payload-induced link deflection.
• Operation near singular configurations.
• Incorrect mapping between world, base, and tool coordinate frames.

These errors can be reduced through calibration, smooth time scaling, appropriate sampling, feedback control, trajectory prevalidation, lower speeds, and compensation for known offsets or backlash. Experimental path measurements should be used to separate modeling errors from controller tracking errors.

Cubic time scaling:

• Enforces specified endpoint positions and zero endpoint velocities.
• Is simple to compute.
• May have nonzero endpoint acceleration.

Quintic time scaling:

• Enforces endpoint position, velocity, and acceleration conditions.
• Produces smoother starts and stops.
• Requires a higher-order polynomial and slightly more computation.

Trapezoidal velocity profile:

• Contains acceleration, constant-velocity, and deceleration phases.
• Makes maximum velocity and acceleration limits intuitive to specify.
• Has acceleration discontinuities at phase transitions unless smoothing is added.

For laboratory straight-line motion, quintic scaling is often preferred when smooth endpoint behavior is important. A trapezoidal profile is useful when motion-time and actuator limits are central. Cubic scaling provides a simpler compromise for moderate-speed demonstrations.