IS-431: Appendix 1 - Upper Mechanism

Appendix 1: Upper Mechanisms (Elgin)

a) Punch Execution

i) Joint Selection

To determine a tangible function specification for our system, we established 2 key matrices through physical measurements and validations.

Requirement		Description
Striking Speed		The system must be able to match a high tempo striking rate. A measurement was done to evaluate the duration required to execute 5 full swings, mimicking the training tempo during training sessions. It was determined to be five 90-degree strikes in 2.62 seconds, or a single 90-degree swing in 0.26 seconds. This necessitates a high angular velocity of 6.04 rad/s.
Training Stick Length		The desired length of the training stick is 90 cm. This specified length, which was physically evaluated, provides a substantial and realistic reach, allowing the robot to engage the user with an arm span analogous to a human opponent. The mechanism is therefore engineered to efficiently actuate training sticks up to this 90 cm maximum, while also accommodating shorter sticks should a user desire an altered reach for different training purposes.

The fidelity of a real boxing match must be maintained. This dictates that the mechanism must not obstruct critical striking zones (like the liver or head) when in its neutral position. A mechanism positioned too low, for example, would render uppercuts to the celiac plexus impossible, thereby training the boxer to aim at an incorrect, clear target rather than the true one.

Concept 1: Multi-Actuator Rig

The first concept involved the adoption of multiple actuators, one for each desired strike. To cover both the left and right sides, this would require a total of six motors. While this could theoretically provide striking fidelity, the solution was deemed mechanically complex, and high cost. Furthermore, the additional motors and mechanisms would create a bulky profile, likely culminating in the obstruction of the user's striking points.

Benefits	Control System	A command to "jab" merely activates a single, pre-programmed motor. No complex inverse kinematics or coordinated trajectory planning is required.
Benefits	Speed	As each motor is optimized for a single, fixed trajectory, mechanical optimization for speed is straightforward.
Limitations	Mechanical	The design is mechanically complex, with a high part count (6 motors, 6 mounts, 6 arms). This leads to a very large physical footprint and high weight.
	Electrical	The system requires power distribution and driver electronics for six separate high-torque motors, significantly increasing cost, thermal load, and potential points of failure.
	Training Fidelity	The six bulky, stationary mechanisms would obstruct the static striking zones (head, liver, celiac plexus). This violates the core design principle to providing a realistic, unobstructed target.

Concept 2: Windmill Actuator

A second concept attempted to consolidate two strikes into one motor. It utilized a "windmill" design, with a training stick extended on both ends of a central motor. A downward rotation would execute a jab, and an upward rotation would execute an uppercut. Although this design reduced the motor count, its extended component for the uppercut would directly obstruct the liver strike zone when in its default, neutral position. This constituted a critical fidelity failure, as it would make one of the key strike zones inaccessible.

Benefits	Mechanical	The motor count is reduced to two per side, simplifying the mechanical assembly and reducing weight.
Benefits	Electrical	Power and control requirements are reduced, lowering the overall cost.
Limitations	Training Fidelity	For the "uppercut" function to work, the opposite end of the training stick must be pointing downwards in the neutral position. This downward-pointing extension directly obstructs the liver and/or celiac plexus striking zones, making them inaccessible to the user.
Limitations		A pure rotational "windmill" motion does not simulate the trajectory of a human jab or uppercut, which follows a natural arc from the shoulder and elbow. This would train an improper defensive reaction from the user.

Concept 3: 2-Degree-of-Freedom (2DOF) Actuator

The final design selection was a 2-Degrees-of-Freedom (2DOF) mechanism, which was determined to offer the optimal balance of fidelity, complexity, and cost by utilizing only two motors per arm. This 2DOF system is comprised of two primary axes. The first is a pitch axis, which provides vertical rotation and permits the execution of the standard downward jab. The inclusion of a secondary yaw axis, providing horizontal rotation, enables the system to execute both the horizontal hook and the diagonal uppercut.

Benefits	Training Fidelity	The coordinated motion of the two joints can accurately simulate the arc, trajectory, and angle of all human punches (jabs, hooks, uppercuts).
Benefits		The arm is compact. In its neutral "guard" position, the mechanism can be retracted upwards and away, leaving all static striking zones completely clear and unobstructed for the user.
Limitations	Control	This system necessitates a complex control system. To move the noodle tip from a start point to an end point, the controller must solve inverse kinematics equations to calculate the precise, coordinated angles for both the yaw and pitch motors.

Selection Criteria

Table A1 was created to aid the decision making for the appropriate concept for the arm joint selection.

Criterion	Priority (3-1) (High–Medium–Low)	Multi-Actuator	Windmill Actuator	2DOF Actuator
Variation of punches executable	3	5	4	5
Fidelity of punches (Angle, Trajectory)	3	3	3	5
Electrical complexity	2	2	4	3
Mechanical complexity	2	4	4	3
Control Complexity	2	5	4	3
Total Score		46	45	48

Table A1: Decision Matrix for Robot Arm Joint Concept

Concept 3 (2-DOF Actuator) is the only solution that meets the requirements of high training fidelity and zero obstruction of the striking zones. The significant challenge of its high control-system complexity is accepted as a trade-off to achieve a functional and effective training product.

ii) Torque Calculations

Arm System Characteristics

Training Sticks

The training sticks, our target object of rotation, is firstly characterized based on the existing polyethylene foam available to us. We first determine the density of the current foam base on its dimensions and weight.

The specification for the new training sticks is based on the desired reach of the robot.

Angular Speed and RPM

Our target matrix is to be able to match the striking tempo of a coach during a training session. We have recreated a series of five 90° strikes and recorded the total duration of these strikes.

Acceleration

The time it takes for the arms to accelerate to the desired speed is crucial to determining the peak torque. A faster acceleration will result in a larger peak torque. To accommodate the users, we base the acceleration time on the human reaction time to visual stimulus, providing the users sufficient time to react. The average human reaction time is typically 0.25 seconds to a visual stimulus (Crossley, 2021).

Arm System Torque Requirements

Static Torque

For rotation in a vertical plane, the maximum static torque happens as the training sticks passes through the horizontal position (ф = 90). At this point, the gravitational torque is at its maximum.

\[ \tau_{g,\text{max}} = M \cdot g \cdot \frac{L}{2} = (0.0763 \;\text{kg}) \cdot (9.81 \;\text{m/s}^2) \cdot \frac{0.9 \;\text{m}}{2} = 0.337 \;\text{N·m} \]

Acceleration Torque

To determine the acceleration torque required, we first model the training sticks as a uniform rod to calculate its inertia.

\[ I_{\text{end}} = \frac{1}{3} M L^2 \] \[ I_{\text{end}} = \frac{1}{3} (0.0763 \;\text{kg})(0.9 \;\text{m})^2 = 0.0206 \;\text{kg·m}^2 \]

The acceleration torque is the required to produce the angular acceleration α.

\[ \tau_\alpha = I_{\text{end}} \cdot \alpha = (0.0206 \;\text{kg·m}^2) \cdot (24.2 \;\text{rad/s}^2) = 0.50 \;\text{N·m} \]

Drag Torque

During the rotation of the training sticks, it also experiences aerodynamic drag which contributes an opposite torque. To maintain a constant rotation, the motor must apply a continuous torque to oppose the opposing drag. The drag force is determined by the following equation:

\[ F_d = \frac{1}{2} C_d \, \rho_{\text{air}} \, A \, v^2 \]

Where C_d is the drag coefficient, and ρ_air is the density of air. To calculate the resultant torque because of drag, we must consider two other factors. Firstly, the velocity of the rotating pool noodle is not a constant but varies with length from the pivoting point. Secondly, the resultant torque must account for the increasing distance across the pool noodle while still accounting for the increase in velocity. To derive the appropriate equation, we first consider the infinitesimal force along the pool noodle as a function of its length.

\[ \text{Velocity } v(r) = r\omega \] \[ dF_d = \frac{1}{2} C_d \, \rho_{\text{air}} (D \cdot dr)(r\omega)^2 = \frac{1}{2} C_d \, \rho_{\text{air}} \, D \, \omega^2 \, r^2 \, dr \]

This force acts at a distance r from the pivot, so the infinitesimal torque from drag is:

\[ d\tau_{\text{drag}} = r \cdot dF_d = r \left( \frac{1}{2} C_d \, \rho_{\text{air}} \, D \, \omega^2 \, r^2 \, dr \right) = \frac{1}{2} C_d \, \rho_{\text{air}} \, D \, \omega^2 \, r^3 \, dr \]

If we integrate from the pivot to the total length of the training sticks, we can determine the equation to calculate the drag torque.

\[ \tau_{\text{drag}} = \int_0^L \frac{1}{2} C_d \, \rho_{\text{air}} \, D \, \omega^2 \, r^3 \, dr = \frac{1}{8} C_d \, \rho_{\text{air}} \, D \, \omega^2 \, L^4 \]

The coefficient of drag is also not a constant and is dependent on the Reynolds number. We account for the highest Reynolds number which occurs at the tip speed, assuming that there is no wind present.

\[ Re = \frac{v \, D}{\nu_{\text{air}}} \]

Where ν_air is the kinematic viscosity of air. [Reference]

The drag coefficient C_d has been defined to be the drag of a circular cylinder in sub-critical flow regime with a value of 1.2. [Reference]

Peak vs. Continuous Torque Requirements

Based on the calculations, the system has two distinct torque requirements.

Peak torque is the maximum, intermittent torque required by the system. It occurs only during the 0.25-second acceleration phase while the arm is simultaneously lifting against the maximum gravitational load (at the horizontal position).

\[ \tau_{\text{peak}} = \tau_\alpha + \tau_{g,\text{max}} \] \[ \tau_{\text{peak}} = 0.50 \;\text{N·m} + 0.337 \;\text{N·m} = 0.837 \;\text{N·m} \]

Continuous torque is the torque required to maintain the arm's operation after acceleration is complete. This load is highest when the arm is moving through the horizontal position at full speed, where the motor must fight both gravity and aerodynamic drag.

\[ \tau_{\text{continuous}} = \tau_{g,\text{max}} + \tau_{\text{drag}} \] \[ \tau_{\text{continuous}} = 0.337 \;\text{N·m} + 0.215 \;\text{N·m} = 0.552 \;\text{N·m} \]

Final Arm System Specifications

It is standard engineering practice to apply a safety margin to account for modelling inaccuracies, unforeseen friction, variations in material properties, and voltage fluctuations. A safety factor of 1.5 is applied to the required torque.

\[ \tau_{\text{peak}} \times 1.5 = 0.837 \;\text{N·m} \times 1.5 = 1.26 \;\text{N·m} \] \[ \tau_{\text{continuous}} \times 1.5 = 0.552 \;\text{N·m} \times 1.5 = 0.828 \;\text{N·m} \]

iii) Motor Selection Criteria

Table A8 was created to aid the decision making for the appropriate motor for the arm joints.

Criterion	Priority (3-1) (High–Medium–Low)	Stepper Motor	Brushed DC Servo	BLDC Servo
Torque at required speed	3	5	3	5
Positioning accuracy & repeatability	3	3	2	5
Disturbance rejection / stiffness	3	1	3	5
Integration complexity (mechanical & electrical)	2	4	4	2
Cost & availability	2	5	3	1
Tuning & commissioning effort	1	4	3	1
Total Score		49	41	52

Table A8: Decision Matrix for Motor Selection