Mechanism and Control of a One-Actuator Mobile Robot Incorporating a Torque Limiter

Abstract

This study presents a novel mechanism for the control of a wheeled robot maneuvering with a single actuator. Elements of snakeboard and two-wheeled skateboard propulsion are applied to the design. Two passive wheels, i.e., casters whose orientation can be controlled, are attached in the back and front of the robot body, and a rotor rotates above the body to induce body propulsion using its counter torque. Three degrees of freedom of motion, i.e., the orientation of the rotor and each of the two casters, are mechanically coupled to the single actuator via a torque limiter. The stoppers are set to restrict the angle of the caster orientation, and a torque limiter allows the rotor to continue rotating without being affected by the stopper’s restriction to the range of motion. Experiments demonstrate that the sinusoidal rotor rotation can propel this robot forward and that adding the increasing or decreasing offset to the sinusoidal rotor rotation can curve the robot’s motion. Next, a method to position the robot at a specified goal position is proposed, assuming that the current position of the robot is detectable in every control cycle. This method adjusts the rate of increase or decrease of the offset in sinusoidal rotor rotation depending on the direction of the goal position. Introducing the motion capture system enables the robot to successfully reach the specified goal positions.

Appendix

1.1 A1 Motion Equation

The position and the orientation of the man body, the front and the rear wheel are denoted by (X₀, Y₀, ϕ₀), (X₁, Y₁, ϕ₁) and (X₂, Y₂, ϕ₂), respectively. The positional relationship among them is represented as follows:

$$ X_{1} = X_{0} + L_{1} \cos \phi_{0} $$

(11)

$$ Y_{1} = Y_{0} + L_{1} \sin \phi_{0} $$

(12)

$$ X_{2} = X_{0} - L_{2} \cos \phi_{0} $$

(13)

$$ Y_{2} = Y_{0} - L_{2} \sin \phi_{0} $$

(14)

Here, L₁ and L₂ are the distance from the center of the main body to each wheel. The velocity of the front and rear wheel normal to the wheel axis,V₁ and V₂, are given as follows:

$$ V_{i} = \dot{X}_{i} \cos \phi + \dot{Y}_{i} \sin \phi $$

(15)

while the constraints that keep the wheels from slipping to each wheel axis direction are written as follows:

$$ V_{i}^{\perp} \equiv \dot{X}_{i} \sin \phi - \dot{Y}_{i} \cos \phi $$

(16)

where, i = 1, 2 distinguishes the front and rear wheels.

The rotor, the torque limiter and the wheels move together with the main body. The main body dynamics is obtained as the following motion equation:

$$ \mathrm{M} \ddot{\mathrm{Q}} = \mathrm{J}_{C}^{T} \mathrm{F}_{C} + \mathrm{J}_{V}^{T} \mathrm{F}_{V} + \mathrm{J}^{T}_{U} u + \mathrm{F}_{f} $$

(17)

Here, $\mathrm {Q} = \left [\begin {array}{ccc} X_{0} & Y_{0} & \phi _{0} \end {array} \right ]^{T}$ is the state vector of the main body, M is the inertial matrix whose components are given by the total mass of the main body including the rotor and wheels M₀, and the moment of inertial of the main bodyI₀ (without rotor),

$$ \mathrm{M} = \left[\begin{array}{ccc} M_{0} & 0 & 0 \\ 0 & M_{0} & 0 \\ 0 & 0 & I_{0} \end{array} \right] $$

(18)

J_C and J_V are the Jacobian matrices

$$ \mathrm{J}_{C} = \left[\begin{array}{crr} \sin \phi_{1} & -\cos \phi_{1} & -L_{1} \cos(\phi_{1}-\phi_{0}) \\ \sin \phi_{2} & -\cos \phi_{2} & L_{2} \cos(\phi_{2}-\phi_{0}) \end{array} \right] $$

(19)

$$ \mathrm{J}_{V} = \left[\begin{array}{ccr} \cos \phi_{1} & \sin \phi_{1} & L_{1} \sin(\phi_{1}-\phi_{0}) \\ \cos \phi_{2} & \sin \phi_{2} & -L_{2} \sin(\phi_{2}-\phi_{0}) \end{array} \right] $$

(20)

that relates $\dot {Q}$ to

$$ \mathrm{V}^{\perp}=[\begin{array}{cc} V_{1}^{\perp} & V_{2}^{\perp} \end{array}]^{T} $$

(21)

$$ \mathrm{V}=[\begin{array}{cc} V_{1} & V_{2} \end{array}]^{T} $$

(22)

respectively, and $\mathrm {F}_{C}= \left [\begin {array}{cc} F_{C1} & F_{C2} \end {array} \right ]^{T}$ is the constraint force that prevents each wheel from slipping to the wheel axis direction, $\mathrm {F}_{V} = \left [\begin {array}{cc} F_{V1} & F_{V2} \end {array} \right ]^{T}$ is the resistance force against the wheel rotation or progression such as the friction. F_f is the friction force, given by the following equation:

$$ \mathrm{F}_{f}=[\begin{array}{ccc} 0 & 0 & \xi_{R} \dot\theta_{R} \end{array}]^{T} $$

(23)

See the Section A for θ_R and ξ_R. J_U becomes as follws:

$$ \mathrm{J}_{U} = \left[\begin{array}{ccc} 0 & 0 & 1 \end{array} \right] $$

(24)

And, u is control input, i.e., the counter force against the rotor rotation u = −τ.

1.2 A2 Velocity Constraints

The wheel is assumed not to slip in its wheel-axis direction. This condition is equivalent to V^⊥ == 0, which is written using the derivative of (11)–(14) as follows:

$$ \mathrm{J}_{C} \dot{\mathrm{Q}} = 0 $$

(25)

1.3 A3 Resistance Force

As the resistance force applied to the wheel motion, the viscous friction is considered. Then, F_V is represented as follows:

$$ \mathrm{F}_{V} = - \mathrm{B} \mathrm{V} = - \mathrm{B} \mathrm{J}_{V} \dot{\mathrm{Q}} $$

(26)

where, B is the viscous matrix:

$$ \mathrm{B} = \left[\begin{array}{cc} b_{1} & 0 \\ 0 & b_{2} \end{array} \right] $$

(27)

b₁ and b₂ are the viscous coefficients corresponding to each wheel.

1.4 A4 Stopper

The collision and contact between the bar and stoppers are modeled as the spring and the dumper, which generates the reaction torque τ_stopper around the wheel rotation.

$$ \tau_{stopper}(\theta) = \left\{ \begin{array}{cl} - b_{s} \dot{\theta} + k_{s} (\theta_{+} - \theta) & (\theta> \theta_{+})\\ 0 & (\theta_{-} < \theta_{M} < \theta_{+}) \\ - b_{s} \dot{\theta} + k_{s} (\theta_{-} - \theta) & (\theta< \theta_{-}) \end{array} \right. $$

(28)

1.5 A5 Coupling Dynamics of Wheel Orientations

Put the orientation angle of the front and the rear wheel relative to the main body as θ₁ = ϕ₁ −ϕ₀ and θ₂ = ϕ₂ −ϕ₀, respectively, and let the torque limiter angle relative to the main body θ₃. Their dynamics are given as follows:

$$ I_{1} \ddot\theta_{1} = - \xi_{1} \dot\theta_{1} + \tau_{1} $$

(29)

$$ I_{2} \ddot\theta_{2} = - \xi_{2} \dot\theta_{2} + \tau_{2} $$

(30)

$$ I_{3} \ddot\theta_{3} = - \xi_{3} \dot\theta_{3} + \tau_{3} $$

(31)

Here, ξ₁, ξ₂, ξ₃ are the viscous coefficients of the rotation around the front wheel, the rear wheel, and the rotor, and τ₁, τ₂ and τ₃ are the torque around them, respectively. The coupling of the wheel orientation imposes the following constraints:

$$ \theta_{1} = \gamma_{1} \theta, \theta_{2} = \gamma_{2} \theta, \theta_{3}=\theta $$

(32)

$$ \tau_{1} + \tau_{2} + \tau_{3} = \tau_{w} $$

(33)

Besides, the stopper limits the range of the rotation, and the torque generated by the motor is transmitted via the torque limiter. Considering all of them, the rotation of the wheel orientation is solely represented by the following dynamics of θ because of the coupling (33)

$$ \begin{array}{@{}rcl@{}} (|\gamma_{1}|I_{1} + |\gamma_{2}|I_{2} + I_{3}) \ddot{\theta} = &-&(|\gamma_{1}| \xi_{1} + |\gamma_{2}| \xi_{2} + \xi_{3}) \dot \theta\\ &+& \tau_{stopper} + \tau_{limiter}(\tau_{w}) \end{array} $$

(34)

1.6 A6 Rotor

The motion of the rotor is simply modeled as the rotation of the rigid bar.

$$ I_{R} \ddot\theta_{R} = - \xi_{R} \dot\theta_{R} + \tau_{R} $$

(35)

Here θ_R is the rotor angle relative to the main body, I_R is the moment of inertia of the rotor, ξ_R is the friction coefficient of the rotor rotation, and τ_R denotes the driving moment.

1.7 A7 Actual Computation

The degrees of freedom of this system becomes five, i.e., X₀, Y₀, ϕ₀, θ and θ_R. The state of the main body is represented by X₀, Y₀ and ϕ₀, whose dynamics are given by (17) with the velocity constraint (16), and is driven as the torque −τ.

Then, its reaction forceτ rotates the rotor as well as the wheel orientation. Namely,

$$ \tau = \tau_{R} + \tau_{w} $$

(36)

However, the dynamics of the rotor and the wheel orientation varies depending on whether the torque limiter is on or off.

If the torque limiter is off, the wheel orientation changes with the rotor rotation and the coupling relation

$$ \dot\theta = \gamma \dot\theta_{R} $$

(37)

holds. Accordingly, (34) and (35) should be solved under this condition which allows us to get $\ddot {\theta } (= \ddot \theta _{R})$. Now, we can obtain the value of the τ_limiter(τ_w) from this $\ddot {\theta }$, using (34). Here,

• a. The case τ_min ≤ τ_limiter (τ_w) ≤ τ_max: The torque limiter is actually off. Thus, we should continue the calculation without any modification.

• b. The case τ_limiter (τ_w) < τ_min: The torque limiter should disconnect the torque transmission. Thus, put τ_w = τ_min and calculate (34) and (35) again removing condition (37).

• c. The case τ_limiter (τ_w) > τ_max: The torque limiter should disconnect the torque transmission. Thus, put τ_w = τ_max and calculate (34) and (35) again removing condition (27).

Finally, we should mention the calculation for the moment at which the torque limiter switches from on to off. In this moment, the two different rotations, $\dot \theta$ and $\dot \theta _{R}$, are coupled and starts to rotate with the same speed $\dot \theta _{new}$. This $\dot \theta _{new}$ is calculated considering the conservation of the angular moment:

$$ (|\gamma_{1}|I_{1} + |\gamma_{2}|I_{2} + I_{3}) \dot{\theta} + I_{R} \dot\theta_{R} {\kern1.7pt}={\kern1.7pt} (|\gamma_{1}|I_{1} + |\gamma_{2}|I_{2} + I_{3} + I_{R}) \dot{\theta}_{new} $$

(38)

Namely, the rotation speed jumps at this moment.

1.8 A8 Default Parameters

The Rung-Kutta method with 0.0001 second step size was utilized for the simulations. The followings are default parameter values in the simulations when any other descriptions are now shown: M = 4, I₀ = 0.08, I₁ = I₂ = 0.002, I₃ = 0.001, I_R = 0.05, L₁ = L₂ = 0.1, γ₁ = 1, γ₂ = − 1, γ = 1, ξ₀ = 0.01, ξ₁ = ξ₂ = ξ = 0, ξ_R = 0.1, b₁ = b₂ = − 3, b_s = 0.2, k_s = 5, θ_L+ = +α, θ_L− = −α. τ_min = − 0.1, τ_max = 0.1,