Assignment 3: Advanced Robotics Assessment

Objective

The goal of this project is to evaluate a 3R articulated robot that moves from a start position to an end position along a desired path within an 11-second time frame. Torque values for all three joint motors will be calculated. For this project, all arms are assumed mass-less.

Arm Length Specifications

• L1 has a constant length of 0.5 m.

• L3 is the third arm and equals:

  L3=Student ID100 (in meters)L3 = \frac{\text{Student ID}}{100} \text{ (in meters)}L3=100Student ID (in meters)

  Example: For ID s1234567,

  L3=0.28 mL3 = 0.28 \text{ m}L3=0.28 m

  It is preferred that L2 and L3 be around 1 m.

• L2 = 2 − L3

• Constraint:

  0.8If not, adjust student number.

Path End Points

A rest-to-rest path is required between the initial and final points:

• Initial point:

  P1=[(L1+L3), 0.5, 0.5]P_1 = [(L1 + L3),\ 0.5,\ 0.5]P1=[(L1+L3),0.5,0.5]

• Final point:

  P2=[−(L1+L3), 0.6, 1.0]P_2 = [-(L1 + L3),\ 0.6,\ 1.0]P2=[−(L1+L3),0.6,1.0]

Forbidden Zone for Robot Tip Point

The robot tip must not enter or touch the following sphere:

(X−0.55)2+(Y−0.75)2+(Z−0.05)2=2(X - 0.55)^2 + (Y - 0.75)^2 + (Z - 0.05)^2 = 2(X−0.55)2+(Y−0.75)2+(Z−0.05)2=2

Mass and Inertia

• m3 = 5 kg – mass of motor at last joint

• m4 = 5 kg – mass at tip (payload)

• All arms are mass-less

• Other motors weigh 5 kg each, if needed

Analysis Requirements

1. Geometric Path Design (5 marks)

Design one fully defined geometric path.

2. Time-Dependent Path Design (5 marks)

Construct a trajectory ensuring:

• Zero velocity at start and end

• Zero acceleration at start and end

• Zero jerk at start and end

3. End-Effector Velocity (5 marks)

Calculate and plot via forward kinematics.

4. End-Effector Acceleration (5 marks)

Calculate and plot.

5. End-Effector Jerk (5 marks)

Calculate and plot.
(Check curves for compatibility.)

6. Joint Coordinates (10 marks)

Use inverse kinematics.

7. Joint Velocities (10 marks)

Use Jacobian and IK.

8. Joint Accelerations (10 marks)

Use Jacobian and IK.

9. Joint Jerk (10 marks)

Use Jacobian and IK.

10. Equations of Motion (10 marks)

Derive dynamic equations for 3R robot.

11. Motor Torques (10 marks)

Compute torque for each joint motor.

12. Motor Power (15 marks)

Compute as:

Power=τ⋅ω\text{Power} = \tau \cdot \omegaPower=τ⋅ω

All outputs must be functions of time and plotted.
Robot cannot go underground.

Design Section

13. Use Calculated Torques to Simulate Joint Variables 

Numerically solve equations of motion.

14. Double the Payload – m4 = 10 kg 

Use same torque values and re-simulate.

15. Compare Joint Variable Time Histories 

Compare outputs from steps 13 and 14.

16. Determine End-Effector Paths

Compare ideal path vs. path with extra payload.

17. Apply Proportional Control 

• Implement proportional controller

• Show effect on error and path

• Gain is selected via trial and error

18. Determine Power Needed for Error Compensation 

Compute power needed for the controlled system.

Brief Summary of Assessment Requirements

Task: Model and analyse a 3R articulated robot that moves from a start to an end point along a rest-to-rest trajectory in 11 seconds, then compute joint torques and motor powers. Arms are massless; specified masses at joint 3 and payload.

Key data & constraints

• L1 = 0.5 m.

• L3 = (StudentID)/100 m → ensure 0.8 < L3 xss=removed>

• Start point P1=[(L1+L3),  0.5,  0.5]P_1=[(L_1+L_3),\;0.5,\;0.5]P1=[(L1+L3),0.5,0.5].

• End point P2=[−(L1+L3),  0.6,  1.0]P_2=[-(L_1+L_3),\;0.6,\;1.0]P2=[−(L1+L3),0.6,1.0].

• Forbidden sphere for tip: (X−0.55)2+(Y−0.75)2+(Z−0.05)2=2(X-0.55)^2+(Y-0.75)^2+(Z-0.05)^2=2(X−0.55)2+(Y−0.75)2+(Z−0.05)2=2 — tip must not enter/touch.

• Masses: m3 = 5 kg (motor at joint 3), m4 = 5 kg (payload). Other motors 5 kg if needed.

• No part of the robot may go underground (z≥0).

Deliverables / analysis steps (mandatory, with marks)

1. Geometric path (define shape).

2. Time-parametrised rest-to-rest trajectory with zero velocity, acceleration and jerk at endpoints.
   3–5. End-effector velocity, acceleration, jerk (forward kinematics) — plots and compatibility checks.
   6–9. Joint coordinates, velocities, accelerations, jerks (inverse kinematics / Jacobian) — time functions and plots.

3. Equations of motion (dynamic model for 3R manipulator).

4. Joint torques (time functions).

5. Motor power (τ·ω, time functions).
   Optional design (extra credit): numerical simulation using computed torques, repeat with doubled payload, compare histories and paths, apply proportional control and compute power needed to compensate error.

How the Academic Mentor Guided the Student 

1. Problem translation & parameter check
We began by confirming L3 from the student ID and enforced the constraint 0.8–1.2 m; adjusted example values so L2≈L3≈1 m. We also checked the start/end coordinates and the forbidden sphere to ensure the geometric path can be drawn without violations.

2. Geometric path design (Step 1)
Mentor and student selected a feasible 3D path (e.g., a smooth cubic spline in space or a circular arc blended with straight segments) that connects P1P_1P1 to P2P_2P2 and skirts the forbidden sphere. The mentor insisted on an analytic parametric form so derivatives are exact.

3. Time-dependent rest-to-rest trajectory (Step 2)
We parametrised motion over t∈[0,11]t\in[0,11]t∈[0,11] s using a 7th-degree polynomial or minimum-jerk quintic whose coefficients ensure position, velocity, acceleration and jerk constraints at t=0t=0t=0 and t=11t=11t=11. Mentor provided the standard boundary conditions and formulas and validated endpoint constraints symbolically and numerically.

4. Forward kinematics & end-effector derivatives (Steps 3–5)
Using the robot’s Denavit–Hartenberg (DH) or geometric model, the mentor guided derivation of x(t), y(t), z(t) and then analytical differentiation (or high-precision numerical differentiation) to obtain velocity, acceleration and jerk. Plots were generated to check smoothness and that no curve violated physical limits or the forbidden zone.

5. Inverse kinematics & Jacobian (Steps 6–9)
The mentor taught an IK solution (closed-form for 3R planar/3D configuration as applicable) to compute joint angles θ1(t), θ2(t), θ3(t). We used the Jacobian J(θ)J(\theta)J(θ) to transform end-effector velocity→joint velocities and compute joint accelerations and jerks with θ˙=J−1x˙\dot\theta = J^{-1}\dot{x}θ˙=J−1x˙ and appropriate differentiation (and Jacobian derivative) for accelerations and jerk. Mentor emphasised numerical stability checks for near-singular configurations.

6. Dynamics: equations of motion (Step 10)
We set up the manipulator dynamics via Lagrangian or Newton–Euler formulations. Mentor demonstrated grouping inertia, Coriolis, centrifugal and gravity (payload and motor masses) terms, simplifying because arms are massless. Symbolic derivation produced: M(θ)θ¨+C(θ,θ˙)θ˙+G(θ)=τM(\theta)\ddot\theta + C(\theta,\dot\theta)\dot\theta + G(\theta) = \tauM(θ)θ¨+C(θ,θ˙)θ˙+G(θ)=τ.

7. Compute torques and power (Steps 11–12)
Substitute θ(t), θ˙(t)\dot\theta(t)θ˙(t), θ¨(t)\ddot\theta(t)θ¨(t) into the dynamic equations to get τ(t). Motor power was computed as Pi(t)=τi(t)⋅ωi(t)P_i(t)=\tau_i(t)\cdot\omega_i(t)Pi(t)=τi(t)⋅ωi(t) (ω = θ˙\dot\thetaθ˙). Mentor advised unit checks and plotted τ(t) and P(t) for all joints.

8. Optional numerical simulation & control (Steps 13–18)
For the design extension, mentor guided a forward numerical integration of the dynamics (using the torques computed as feedforward inputs) to obtain θ_sim(t). Then m4 was doubled and re-simulated, producing comparative plots. A proportional controller τctrl=Kp(θref−θ) \tau_{ctrl} = K_p(\theta_{ref}-\theta)τctrl=Kp(θref−θ) was implemented; mentor helped tune Kp by simulation to reduce tracking error and evaluated additional power required for error compensation.

9. Verification & reporting
Mentor checked forbidden-zone avoidance, z-nonnegativity, energy plausibility, and curve continuity. All plots were annotated, and assumptions documented (massless arms, frictionless joints unless included).

Final Outcome and How It Was Achieved

What the student delivered

• A defined geometric path and an 11 s time law satisfying rest-to-rest (zero v,a,jerk at endpoints).

• Time series and labelled plots for end-effector position, velocity, acceleration, jerk.

• Inverse kinematics solution and joint coordinate time histories; joint velocities, accelerations, jerks computed via Jacobian and validated against forward derivatives.

• Dynamic model M(θ),C(θ,θ˙),G(θ)M(\theta),C(\theta,\dot\theta),G(\theta)M(θ),C(θ,θ˙),G(θ) and time-domain τ(t) for each joint.

• Motor power profiles Pi(t)P_i(t)Pi(t) and peak/average power values.

• Optional simulations: forward numerical integration using τ(t); re-run with m4 doubled; comparative plots showing greater deviation when payload increased; P-controller implementation and plots showing reduced tracking error and required compensation power.

How it was produced (tools & methods)

• Symbolic algebra and numeric computation in MATLAB / Python (NumPy, SciPy, SymPy) for derivations and simulations.

• High-resolution time discretisation for numerical stability.

• Plots annotated for all required quantities and exported for the report.

Learning Objectives Covered

• Kinematic modelling (forward and inverse).

• Use of Jacobian for velocity/acceleration mapping and singularity awareness.

• Trajectory planning with kinematic boundary constraints (position, velocity, acceleration, jerk).

• Dynamic modelling (Lagrange/Newton–Euler) and torque computation.

• Power estimation for actuators and energy considerations.

• Numerical simulation of equations of motion and payload sensitivity analysis.

• Basic feedback control (proportional) design and performance evaluation.

• Engineering reporting: assumptions, unit consistency, validation, and graphical communication.

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