Knee development

This is an early draft, need to define the weight of pib, motor torques and speeds

In this document, we outline the features and capabilities required for the knee of pib, our humanoid robot. By understanding these requirements, we can determine the appropriate transmission ratio and method of transmission to ensure effective movement and stability.

Movement Requirements

The knee should be able to achieve a bending angle of at least 135°. This range of motion is critical for the robot's ability to perform everyday activities, such as squatting, standing up, and walking with stability and balance.

Torque Calculation

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The knee experiences its maximum static torque when pib is in the lowest possible squatting position. Assuming that the knee can bend to the maximum angle of 135°, the torque in this position can be calculated using the following formula:

T_stat = F * x

Where:

• F = Force exerted due to the robot's weight

• x = Distance from the axis of rotation (lever arm length)

Let's assume:

• The mass of pib is 40 kg

• Gravitational acceleration g is 9.81 m/s²

Thus:

F = 40 kg * 9.81 m/s² = 392.4 N
x = 0.370 m
T_stat = 392.4 N * 0.370 m = 145.2 Nm

Since there are two knees, the torque per knee would be 145.2 Nm / 2 = 72.6 Nm for each knee.

To determine the maximum torque the knee should be capable of handling, we consider the scenario where pib is balancing entirely on one knee. In addition, a safety margin must be included to account for dynamic loads and other uncertainties. Let’s assume a safety margin factor of 3.0, which leads to a maximum dynamic torque:

T_dyn, max = 145.2 Nm * 3.0 = 435.6 Nm

Movement Speed

Assuming that pib should be able to stand up from a squatting position within 1 second, we need to calculate the necessary rotational speed for the knee joint. This can be approximated by considering the angular displacement over time. The knee joint should be able to rotate from 135° to 0° in 1 second. We denote the rotational speed as ω (omega) and calculate it as follows:

ω = Θ / t

Where:

• Θ is the angular displacement (135° or 2.36 radians)

• t is the time (1 second)

Thus:

ω = 2.36 rad/s

We assume that the dynamic speed for balancing is the same as the "static" movements.

Transmission Ratio

The motor alone cannot generate such a high torque directly; therefore, a transmission mechanism is required to increase the torque output. However, increasing torque via transmission comes with a trade-off: it reduces the rotational speed of the knee joint proportionally to the transmission ratio.

To match the required dynamic torque of 435.6 Nm, we will select a suitable transmission ratio r such that:

T_motor * r = T_dyn, max

Where T_motor is the torque output of the motor. Depending on the motor specifications, we will choose a transmission ratio that balances both torque requirements and movement speed to achieve the desired performance for the knee joint.

2-Stage Transmission

To achieve a transmission of 36 as assumed in the table below, 2 stages seem to be useful:

1. Planetary gear-stage with ration 1:7

2. belt-drive with 1:6.25 (75/12 teeth)

Summary Table of Movement Speeds and Torques