Control of stance-leg motion and zero-moment point for achieving perfect upright stationary state of rimless wheel type walker with parallel linkage legs

Abstract

The authors have studied models and control methods for legged robots without having active ankle joints that can not only walk efficiently but also stop and developed a method for generating a gait that starts from an upright stationary state and returns to the same state in one step for a simple walker with one control input. It was clarified, however, that achieving a perfect upright stationary state including zero dynamics is impossible. Based on the observation, in this paper we propose a novel robotic walker with parallel linkage legs that can return to a perfect stationary standing posture in one step while simultaneously controlling the stance-leg motion and zero-moment point (ZMP) using two control inputs. First, we introduce a model of a planar walker that consists of two eight-legged rimless wheels, a body frame, a reaction wheel, and massless rods and describe the system dynamics. Second, we consider two target control conditions; one is control of the stance-leg motion, and the other is control of the ZMP to stabilize zero dynamics. We then determine the control input based on the two conditions with the target control period derived from the linearized model and consider adding a sinusoidal control input with an offset to correct the resultant terminal state of the reaction wheel. The validity of the proposed method is investigated through numerical simulations.

1. Introduction

Limit cycle walkers generate forward motion from one impact posture to the next by continuously applying minimal control input so as not to fail to overcome the potential barrier at mid-stance [1–3]. Since this approach takes full advantage of the natural dynamics inherent in the robot’s body, some problems such as the inability to stop walking once started and the difficulty in finding an appropriate initial state arise in return for the high efficiency. In summary, the greatest weaknesses of limit cycle walkers are the inability to stop their motion at any given moment and that to remain stationary on one leg.

To solve the above problem, the authors started the study of motion generation method for being able to start from the origin on the phase plane and return to the same point in one step. First, we introduced a simple 1-input 2-DOF model that consists of a rimless wheel (RW) and a reaction wheel, called the underactuated rimless wheel (URW) [4–7], and developed a method for generating the desired one-step motion [8]. In this method, by setting the time-integrated value of the control input to zero, it was possible to appropriately control the stance-leg motion and achieve limit cycle stability of the reaction wheel simultaneously. It was impossible, however, to control the angular velocity of the reaction wheel to zero simultaneously while standing still, and we concluded that point-footed walkers only with one control input cannot achieve a perfect stationary state or stand still with zero kinetic energy. After that, this issue remained unresolved.

Since RWs with symmetrical shapes have a constant impact posture, they always generate an asymptotically stable, period-1 passive-dynamic gait on a gentle downhill [9]. Active RW-type walkers, which inherit this feature, have the great advantage of being able to adapt to various terrains with a simple control method that does not require swing-leg control. Due to the point-foot structure, however, it is impossible to create a perfect stationary state as mentioned above. If we consider a simple-legged robot model in which the flat foot and stance leg are connected via an active ankle joint, it is clear that this model can achieve a perfect stationary state. For this, however, it is necessary to apply the ankle-joint torque while fitting the flat foot to the floor. If we try to achieve this with a RW-type walker, flat feet must be attached to the end of each leg frame via an active ankle joint [10]. As a result, the weight of the robot significantly increases and the control system becomes complicated. To solve this difficulty and achieve the target motion with as few motors as possible, a novel RW-type walker must be devised.

Based on the observation, in this paper we propose a novel RW-type walker formed by four parallel linkage legs, a body frame, a reaction wheel, and massless rods as shown in Figure 1 top. In other words, this is the same as a combined rimless wheel (CRW) that consists of two identical eight-legged RWs [11, 12] with a reaction wheel on the short body frame. The most significant feature of the model is that the leg link does not have an active joint at the ankle, but the stance-leg motion and zero-moment point (ZMP) [13, 14] can be controlled independently by using two control inputs. Until now, various studies on closed-link mechanisms for robot legs have been done, and the purpose has also been various, and many of them have been aimed at creating complex movements with a single or a small number of driving forces based on kinematics [15–19]. In general, bipeds whose legs are composed of serial links, there is the difficulty of simultaneously controlling the stance-leg motion and ZMP using the ankle-joint torque. Our robot in Figure 1, on the other hand, has a novelty capable of separating the above two tasks by using a parallel link mechanism without destroying the simple kinematics and dynamics as a single RW [9, 20, 21]. With the massless rod corresponding to the stance foot fitted to the floor, while preferentially controlling the stance-leg motion to start from a stationary state and return to the same state, the same can be achieved for the reaction wheel motion using the degree of freedom of ZMP.

The subsequent sections are organized as follows. In Section 2, we describe the model assumptions and the equations of motion, constraint conditions, and collision for stance-leg exchange. In Section 3, we design a control system for generating stance-leg motion that completes in one step by dividing into two phases; one is the phase in which the stance leg rotates forward from an upright stationary state with constant angular acceleration, and the other is the phase in which it returns to the upright stationary state again using discrete-time output deadbeat control (DODC) [4, 6] after stance-leg exchange. In Section 4, we derive an approximate analytical solution of the target control period of the first phase using a linearized model [8] and propose a method for correcting the resultant terminal state of the reaction wheel to the origin on the phase plane. In Section 5, we discuss the validity of the proposed method through numerical simulations of the nonlinear model.

Figure 1. Model of planar robotic walker that consists of two identical eight-legged rimless wheels, body link, reaction wheel, and massless rods.

2. Modeling

2.1. Model assumptions

Figure 1 top shows the overview of the RW-type walker. This model is the same as the CRW, in which two identical eight-legged RWs with a symmetrical shape are combined via a rigid body frame. The relative angles between the two adjacent leg frames are $\alpha = \frac{\pi }{4}$ [rad], and the end positions of the rear and foreleg frames are connected by massless rods to always perfectly synchronize the absolute leg angles. In the following, the parallelogram link in contact with the ground, filled with pink in the figure, is called the stance leg, the lower massless rod of it touching the ground is called the sole, and the ground-contact points of the rear/fore RW are called the rear/fore foot. A reaction wheel is attached to the center of the body frame via an active joint, and this is used to appropriately control the ratio of the vertical ground reaction forces acting at the rear and forefeet.

Figure 1 bottom left shows the coordinate system of the robot. The coordinate system has $X$ -axis horizontal forward and $Z$ -axis vertical up with the origin fixed to the ground. $(x,z)$ is the rear foot position and $(x',z')$ is the fore foot position. $\theta _1$ and $\theta _3$ are the absolute angles of the rear and fore RWs relative to the vertical up direction, and $\theta _2$ and $\theta _4$ are those of the body frame and reaction wheel with respect to the horizontal forward direction. The robot has four masses: the mass of the rear RW is $m_1$ , that of the body frame is $m_2$ , that of the fore RW is $m_3$ , and that of the reaction wheel is $m_4$ . The inertia moment around $m_j$ is $I_j \ (j \in \{1,2,3,4\})$ . $L_1$ is the radius of the rear RW, $L_2$ is half the length of the body frame, and $L_3$ is the radius of the fore RW. We assume that the two RWs are the same, that is, $m_1 = m_3$ , $I_1 = I_3$ and $L_1 = L_3$ , and that they move in perfect synchronization with each other or $\theta _1 \equiv \theta _3$ and $\dot{\theta }_1 \equiv \dot{\theta }_3$ always hold during motion. In the following, these assumptions are used as necessary in the equations and explanations.

As illustrated in Figure 1 bottom right, the robot has two control inputs; $u_1$ is the rotation torque between the rear RW and body frame used to control the stance-leg motion, and $u_2$ is the rotation torque between the body frame and reaction wheel to control the ratio of rear and fore vertical ground reaction forces, $F_z$ and $F^{\prime}_z$ . Actually, the rear RW is heavier by the amount of the actuator for applying $u_1$ , but the same thing is attached to the fore RW as a counterweight to balance it. Since the control torque $u_1$ can act as a rotation torque to the rear RW around point P or that to the fore RW around P’ as shown in the figure, it would be possible to control the stance leg as 1-DOF robot arm when the ZMP is in the sole or $F_z \gt 0$ and $F^{\prime}_z \gt 0$ .

2.2. Equation of motion, constraint conditions, and reduced dynamics

Let $\boldsymbol{q} = \left [ \begin{array}{cccccc} x & z & \theta _1 & \theta _2 & \theta _3 & \theta _4 \end{array} \right ]^{\textrm{T}}$ be the generalized coordinate vector. The robot equation of motion then becomes

(1) \begin{equation}\boldsymbol{M} \ddot{\boldsymbol{q}} +\boldsymbol{h} =\boldsymbol{S}\boldsymbol{u} +\boldsymbol{J}_c^{\textrm{T}}{\boldsymbol{\lambda }}_c, \end{equation}

where $\boldsymbol{M} \in{{\mathbb{R}}}^{6 \times 6}$ is the inertia matrix, $\boldsymbol{h} \in{{\mathbb{R}}}^6$ is the nonlinear velocity and gravity vector, and $\boldsymbol{S}\boldsymbol{u} \in{{\mathbb{R}}}^6$ is the control input vector. They are detailed as follows.

(2) \begin{eqnarray} &\boldsymbol{M} = \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} m & 0 & m L_1 \cos \theta _1 & -m L_2 \sin \theta _2 & 0 & 0 \\ 0 & m & -m L_1 \sin \theta _1 & -m L_2 \cos \theta _2 & 0 & 0 \\ m L_1 \cos \theta _1 & -m L_1 \sin \theta _1 & I_1 + m L_1^2 & m L_1 L_2 \sin (\theta _1 - \theta _2) & 0 & 0 \\ -m L_2 \sin \theta _2 & -m L_2 \cos \theta _2 & m L_1 L_2 \sin (\theta _1 - \theta _2) & I_2 + m L_2^2 & 0 & 0 \\ 0 & 0 & 0 & 0 & I_3 & 0 \\ 0 & 0 & 0 & 0 & 0 & I_4 \end{array} \right ] & \end{eqnarray}

(3) \begin{eqnarray} &\boldsymbol{h} = \left [ \begin{array}{c} - m \left ( L_1 \dot{\theta }_1^2 \sin \theta _1 + L_2 \dot{\theta }_2^2 \cos \theta _2 \right ) \\ -m \left ( L_1 \dot{\theta }_1^2 \cos \theta _1 - L_2 \dot{\theta }_2^2 \sin \theta _2 - g \right ) \\ -m L_1 \left ( L_2 \dot{\theta }_2^2 \cos ( \theta _1 - \theta _2) + g \sin \theta _1 \right ) \\ m L_2 \left ( L_1 \dot{\theta }_1^2 \cos (\theta _1 - \theta _2) - g \cos \theta _2 \right ) \\ 0 \\ 0 \end{array} \right ], \ \ \boldsymbol{S} = \left [ \begin{array}{c@{\quad}c} 0 & 0 \\ 0 & 0 \\ 1 & 0 \\ -1 & 1 \\ 0 & 0 \\ 0 & -1 \end{array} \right ], \ \ \boldsymbol{u} = \left [ \begin{array}{c} u_1 \\ u_2 \end{array} \right ] & \end{eqnarray}

The velocity constraint conditions that the rear and forefeet are in contact with the ground without sliding are described as

(4) \begin{align} \dot{x} &= 0, \end{align}

(5) \begin{align} \dot{z} &= 0, \end{align}

(6) \begin{align} \dot{x}' &= \frac{\textrm{d}}{\textrm{d}t} \left ( x + L_1 \sin \theta _1 + 2 L_2 \cos \theta _2 - L_3 \sin \theta _3 \right ) = 0, \end{align}

(7) \begin{align} \dot{z}' &= \frac{\textrm{d}}{\textrm{d}t} \left ( z + L_1 \cos \theta _1 - 2 L_2 \sin \theta _2 - L_3 \cos \theta _3 \right ) = 0. \end{align}

Summarizing these four equations becomes

(8) \begin{equation}\boldsymbol{J}_c \dot{\boldsymbol{q}} = \textbf{0}_{4 \times 1}, \end{equation}

where

(9) \begin{equation}\boldsymbol{J}_c = \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & L_1 \cos \theta _1 & -2L_2 \sin \theta _2 & -L_3 \cos \theta _3 & 0 \\ 0 & 1 & -L_1 \sin \theta _1 & -2L_2 \cos \theta _2 & L_3 \sin \theta _3 & 0 \end{array} \right ]. \end{equation}

On a slippery road surface, to perfectly synchronize the rear and fore RWs, we must introduce the condition $\dot{\theta }_1 = \dot{\theta }_3$ derived from the massless rod instead of the conditions $\dot{x} = 0$ and $\dot{x}' = 0$ [22, 23]. On a non-slip road surface, however, this is not necessary because the above four conditions include the physical meaning of it.

By solving Eqs. (1) and (8) for ${\boldsymbol{\lambda }}_c$ , we obtain

(10) \begin{equation}{\boldsymbol{\lambda }}_c = \left [ \begin{array}{c} F_x \\ F_z \\ F^{\prime}_x \\ F^{\prime}_z \end{array} \right ] = -\boldsymbol{X}_c^{-1} \left (\boldsymbol{J}_c\boldsymbol{M}^{-1} \left (\boldsymbol{S}\boldsymbol{u} -\boldsymbol{h} \right ) + \dot{\boldsymbol{J}}_c \dot{\boldsymbol{q}} \right ), \end{equation}

where $\boldsymbol{X}_c \;:\!=\;\boldsymbol{J}_c\boldsymbol{M}^{-1}\boldsymbol{J}_c^{\textrm{T}}$ . By substituting Eq. (10) into Eq. (1) and arranging it, we obtain the simplified robot equation of motion as

(11) \begin{equation}\boldsymbol{M} \ddot{\boldsymbol{q}} =\boldsymbol{Y}_c \left (\boldsymbol{S}\boldsymbol{u} -\boldsymbol{h} \right ) -\boldsymbol{J}_c^{\textrm{T}}\boldsymbol{X}_c^{-1} \dot{\boldsymbol{J}}_c \dot{\boldsymbol{q}}, \end{equation}

where $\boldsymbol{Y}_c \;:\!=\;\boldsymbol{I}_6 -\boldsymbol{J}_c^{\textrm{T}}\boldsymbol{X}_c^{-1}\boldsymbol{J}_c\boldsymbol{M}^{-1}$ .

The robot model has six generalized coordinates, but it has four constraints, so it is essentially 2-DOF, $\theta _1$ and $\theta _4$ . The essential (reduced) dynamics then becomes

(12) \begin{equation} \left [ \begin{array}{c@{\quad}c} I_5 & 0 \\ 0 & I_4 \end{array} \right ] \left [ \begin{array}{c}{\ddot{\theta }}_1 \\{\ddot{\theta }}_4 \end{array} \right ] + \left [ \begin{array}{c} -m g L_1 \sin \theta _1 \\ 0 \end{array} \right ] = \left [ \begin{array}{c} u_1 \\ -u_2 \end{array} \right ], \end{equation}

where $I_5 \;:\!=\; 2 I_1 + m L_1^2$ and $m \;:\!=\; 2 m_1 + m_2 + m_4$ are the robot’s total mass. Eq. (12) seems like a decoupled system, but it is shown to be equivalent to an URW capable of ZMP control through transformation of the control input as described later.

2.3. Collision equation

On the assumption that the rear feet take off immediately after the forefeet collide with the ground, the perfect inelastic collision equation is derived as

(13) \begin{equation}\boldsymbol{M} \dot{\boldsymbol{q}}^+ =\boldsymbol{M} \dot{\boldsymbol{q}}^- +\boldsymbol{J}_I^{\textrm{T}}{\boldsymbol{\lambda }}_I. \end{equation}

Here, the superscripts “ $-$ ” and “ $+$ ” denote immediately before and immediately after impact. The velocity constraint conditions of the forefeet immediately after impact are specified as

(14) \begin{align} \frac{\textrm{d}}{\textrm{d}t} \left ( x + L_1 \sin \theta _1 - L_1 \sin \left ( \theta _1 - \alpha \right ) \right )^+ &= 0, \end{align}

(15) \begin{align} \frac{\textrm{d}}{\textrm{d}t} \left ( z + L_1 \cos \theta _1 - L_1 \cos \left ( \theta _1 - \alpha \right ) \right )^+ &= 0, \end{align}

(16) \begin{align} \frac{\textrm{d}}{\textrm{d}t} \left ( x + 2 L_2 + L_3 \sin \theta _3 - L_3 \sin \left ( \theta _3 - \alpha \right ) \right )^+ &= 0, \end{align}

(17) \begin{align} \frac{\textrm{d}}{\textrm{d}t} \left ( z + L_3 \cos \theta _3 - L_3 \cos \left ( \theta _3 - \alpha \right ) \right )^+ &= 0. \end{align}

Note, however, that since leg exchange is not taken into account here, $\theta _1$ and $\theta _3$ themselves remain at their values immediately before the collision but their time derivatives are the values immediately after the collision. Summarizing the above four equations becomes

(18) \begin{equation}\boldsymbol{J}_I \dot{\boldsymbol{q}}^+ = \textbf{0}_{4 \times 1}, \end{equation}

where

(19) \begin{equation}\boldsymbol{J}_I = \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & L_1 \cos \theta _1^- - L_1 \cos \left ( \theta _1^- - \alpha \right ) & 0 & 0 & 0 \\ 0 & 1 & -L_1 \sin \theta _1^- + L_1 \sin \left ( \theta _1^- - \alpha \right ) & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & L_3 \cos \theta _3^- - L_3 \cos \left ( \theta _3^- - \alpha \right ) & 0 \\ 0 & 1 & 0 & 0 & -L_3 \sin \theta _3^- + L_3 \sin \left ( \theta _3^- - \alpha \right ) & 0 \end{array} \right ]. \end{equation}

By solving Eqs. (13) and (18) for the velocity vector immediately after impact, we obtain

(20) \begin{equation} \dot{\boldsymbol{q}}^+ = \left (\boldsymbol{I}_6 -\boldsymbol{M}^{-1}\boldsymbol{J}_I^{\textrm{T}} \left (\boldsymbol{J}_I\boldsymbol{M}^{-1}\boldsymbol{J}_I^{\textrm{T}} \right )^{-1}\boldsymbol{J}_I \right ) \dot{\boldsymbol{q}}^-. \end{equation}

After this, with the update of the ground-contact point, we must reset the first and second components of $\dot{\boldsymbol{q}}^+$ as $\dot{x}^+ = 0$ and $\dot{z}^+ = 0$ . The update formula for the angular velocity of the stance leg is obtained by extracting the relevant part from Eq. (20) and is described as

(21) \begin{equation} \dot{\theta }_1^+ = \xi \dot{\theta }_1^-, \ \ \xi = \frac{m L_1^2 \cos \alpha + 2 I_1}{m L_1^2 + 2 I_1}. \end{equation}

Here, $\xi$ is a positive constant less than $1$ . Finally, the position coordinates are reset to

(22) \begin{equation} x^+ = x^- + 2 L_1 \sin \frac{\alpha }{2}, \ \ z^+ = 0, \ \ \theta _1^+ = \theta _3^+ = \theta _1^- - \alpha = \theta _3^- - \alpha = - \frac{\alpha }{2}. \end{equation}

3. Control system design

3.1. Target conditions and determination of control input

For achieving the desired walking motion, the stance-leg motion must be generated with the highest priority. First, we set the stance-leg angle to the control output which is described as $\theta _1 = \left [ \begin{array}{cccccc} 0\; & 0\; & 1\; & 0\; & 0\; & 0 \end{array} \right ]\boldsymbol{q} \;=\!:\; \boldsymbol{C}_1\boldsymbol{q}$ . We then consider the following target condition so that the second-order derivative with respect to time is equal to the angular acceleration command signal, $v[i]$ .

(23) \begin{eqnarray}{\ddot{\theta }}_1 &=&\boldsymbol{C}_1 \ddot{\boldsymbol{q}} \equiv v[i] \nonumber \\ &=&\boldsymbol{C}_1\boldsymbol{M}^{-1}\boldsymbol{Y}_c \left (\boldsymbol{S}\boldsymbol{u} -\boldsymbol{h} \right ) -\boldsymbol{C}_1\boldsymbol{M}^{-1}\boldsymbol{J}_c^{\textrm{T}}\boldsymbol{X}_c^{-1} \dot{\boldsymbol{J}}_c \dot{\boldsymbol{q}} \end{eqnarray}

Second, we consider the following condition so that the vertical ground reaction forces of the rear and forelegs are always equal, that is, the ZMP is always positioned at the center point of the sole.

(24) \begin{equation} 0 \equiv F_z - F^{\prime}_z = \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & 1 & 0 & -1 \end{array} \right ]{\boldsymbol{\lambda }}_c \;=\!:\; \boldsymbol{C}_2{\boldsymbol{\lambda }}_c \end{equation}

By substituting Eq. (10) into Eq. (24), we obtain

(25) \begin{equation}\boldsymbol{C}_2\boldsymbol{X}_c^{-1}\boldsymbol{J}_c\boldsymbol{M}^{-1}\boldsymbol{S}\boldsymbol{u} =\boldsymbol{C}_2\boldsymbol{X}_c^{-1} \left (\boldsymbol{J}_c\boldsymbol{M}^{-1}\boldsymbol{h} + \dot{\boldsymbol{J}}_c \dot{\boldsymbol{q}} \right ). \end{equation}

The target condition is then determined as ${\boldsymbol{\Phi }}\boldsymbol{u} ={\boldsymbol{\Gamma }}$ where

(26) \begin{align}{\boldsymbol{\Phi }} &= \left [ \begin{array}{c}\boldsymbol{C}_1\boldsymbol{M}^{-1}\boldsymbol{Y}_c\boldsymbol{S} \\\boldsymbol{C}_2\boldsymbol{X}_c^{-1}\boldsymbol{J}_c\boldsymbol{M}^{-1}\boldsymbol{S} \end{array} \right ] = \left [ \begin{array}{c@{\quad}c} \frac{1}{m L_1^2 + 2 I_1} & 0 \\ -\frac{1}{L_2} & \frac{1}{L_2} \end{array} \right ], \end{align}

(27) \begin{align} {\boldsymbol{\Gamma }} &= \left [ \begin{array}{c} v[i] +\boldsymbol{C}_1\boldsymbol{M}^{-1} \left (\boldsymbol{Y}_c\boldsymbol{h} +\boldsymbol{J}_c^{\textrm{T}}\boldsymbol{X}_c^{-1} \dot{\boldsymbol{J}}_c \dot{\boldsymbol{q}} \right ) \\\boldsymbol{C}_2\boldsymbol{X}_c^{-1} \left (\boldsymbol{J}_c\boldsymbol{M}^{-1}\boldsymbol{h} + \dot{\boldsymbol{J}}_c \dot{\boldsymbol{q}} \right ) \end{array} \right ] = \left [ \begin{array}{c} v[i] - \frac{m g L_1 \sin \theta _1}{mL_1^2 + 2I_1} \\ 0 \end{array} \right ]. \end{align}

The control input solved is then solved as

(28) \begin{equation}\boldsymbol{u} ={\boldsymbol{\Phi }}^{-1}{\boldsymbol{\Gamma }} = \left [ \begin{array}{c} 1 \\ 1 \end{array} \right ] \left ( (mL_1^2 + 2I_1) v[i] - m g L_1 \sin \theta _1 \right ) . \end{equation}

From this, it is immediately clear that $u_1 = u_2$ , but this result is derived from solving ${\boldsymbol{\Phi }}\boldsymbol{u} ={\boldsymbol{\Gamma }}$ so as to satisfy the condition $F_z = F^{\prime}_z$ . We will discuss this issue again in the next section.

3.2. Stance-leg motion generation

3.2.1. Phase I

The first phase starting from a static standing posture until immediately before impact is defined as Phase I. In this phase, the angular acceleration command signal $v$ is set to a positive constant value $v[0]$ , that is, ${\ddot{\theta }}_1 = v [0]$ . The angular velocity and angular position at $t$ then become

(29) \begin{equation} \dot{\theta }_1 (t) = v[0] t, \ \ \theta _1 (t) = \frac{v[0] t^2}{2}. \end{equation}

Let $T_1$ be the target period of this phase. The target terminal angular position in this phase then satisfies

(30) \begin{equation} \theta _1 \left ( T_1^- \right ) = \frac{v[0] T_1^2}{2} = \frac{\alpha }{2}. \end{equation}

By solving Eq. (24) for $v[0]$ , we obtain

(31) \begin{equation} v[0] = \frac{\alpha }{T_1^2}. \end{equation}

The terminal angular velocity in this phase also becomes

(32) \begin{equation} \dot{\theta }_1 \left ( T_1^- \right ) = v[0] T_1 = \frac{\alpha }{T_1}. \end{equation}

3.2.2. Phase II

The second phase starting immediately after impact is defined as Phase II. The angular position and angular velocity of the stance leg immediately after impact are reset to

(33) \begin{align} \theta _1 \left ( T_1^+ \right ) &= \theta _1 \left ( T_1^- \right ) - \alpha = -\frac{\alpha }{2}, \end{align}

(34) \begin{align} \dot{\theta }_1 \left ( T_1^+ \right ) &= \xi \dot{\theta }_1 \left ( T_1^- \right ), \end{align}

and these are the initial condition in this phase. The state-space realization of ${\ddot{\theta }}_1 = v[i]$ becomes

(35) \begin{equation} \frac{\textrm{d}}{\textrm{d}t} \left [ \begin{array}{c} \theta _1 \\ \dot{\theta }_1 \end{array} \right ] = \left [ \begin{array}{c@{\quad}c} 0 & 1 \\ 0 & 0 \end{array} \right ] \left [ \begin{array}{c} \theta _1 \\ \dot{\theta }_1 \end{array} \right ] + \left [ \begin{array}{c} 0 \\ 1 \end{array} \right ] v[i]. \end{equation}

Let $T_2$ be the target period of this phase, and $\theta _1$ be the control output. We then consider DODC [4] for settling $\theta _1$ and $\dot{\theta }_1$ to zero in two steps. The system of Eq. (30) is discretized with the control period to $T_2/2$ as

(36) \begin{equation} \left [ \begin{array}{c} \theta _1[i+1] \\ \dot{\theta }_1[i+1] \end{array} \right ] = \left [ \begin{array}{c@{\quad}c} 1 & \frac{T_2}{2} \\ 0 & 1 \end{array} \right ] \left [ \begin{array}{c} \theta _1[i] \\ \dot{\theta }_1[i] \end{array} \right ] + \left [ \begin{array}{c} \frac{T_2^2}{8} \\ \frac{T_2}{2} \end{array} \right ] v[i]. \end{equation}

We denote this system as $\boldsymbol{x}_s [i+1] =\boldsymbol{A}_s\boldsymbol{x}_s [i] +\boldsymbol{B}_s v[i]$ . Note that the state variables of the discretized system were defined as

(37) \begin{align}\boldsymbol{x}_s [1] &= \left [ \begin{array}{c} \theta _1 \left ( T_1^+ \right ) \\ \dot{\theta }_1 \left ( T_1^+ \right ) \end{array} \right ] = \left [ \begin{array}{c} -\frac{\alpha }{2} \\ \frac{\alpha \xi }{T_1} \end{array} \right ], \end{align}

(38) \begin{align}\boldsymbol{x}_s [i+1] &= \left [ \begin{array}{c} \theta _1 \left ( T_1^+ + i \frac{T_2}{2} \right ) \\ \dot{\theta }_1 \left ( T_1^+ + i \frac{T_2}{2} \right ) \end{array} \right ] \ \ \left ( i \geq 1 \right ). \end{align}

The feedback gain $\boldsymbol{F}$ that assigns all poles of the closed-loop system $\boldsymbol{A}_s +\boldsymbol{B}_s\boldsymbol{F}$ at zero can be obtained as $\boldsymbol{F} = \left [ \begin{array}{cc} -\frac{4}{T_2^2}\; & -\frac{3}{T_2} \end{array} \right ]$ . The angular acceleration command signals in this phase are then determined as

(39) \begin{align} v[1] &=\boldsymbol{F}\boldsymbol{x}_s [1] \ \ \left ( T_1^+ \leq t \lt T_1^+ + \frac{T_2}{2} \right ), \end{align}

(40) \begin{align} v[2] &=\boldsymbol{F}\boldsymbol{x}_s [2] \ \ \left ( T_1^+ + \frac{T_2}{2} \leq t \lt T_1^+ + T_2 \right ). \end{align}

According to the method of DODC, $\boldsymbol{x}_s [3] = \textbf{0}_{2 \times 1}$ is achieved.

4. Linearized reduced model and stabilization of zero dynamics

4.1. Relationship between control inputs and zero-moment point

The difference between the vertical ground reaction force of the rear leg and that of the fore leg satisfies the following relationship.

(41) \begin{equation} F_z - F^{\prime}_z = \frac{u_1 - u_2}{L_2} \end{equation}

Therefore, we can understand that $F_z \equiv F^{\prime}_z$ is equivalent to $u_1 \equiv u_2$ . Considering Eq. (41), the control input vector for the reduced system can be arranged to

(42) \begin{equation} \left [ \begin{array}{c} u_1 \\ -u_2 \end{array} \right ] = \left [ \begin{array}{c} u_1 \\ -u_1 \end{array} \right ] + \left [ \begin{array}{c} 0 \\ u_1 - u_2 \end{array} \right ] = \left [ \begin{array}{c} 1 \\ -1 \end{array} \right ] u_1 + \left [ \begin{array}{c} 0 \\ \left ( F_z - F^{\prime}_z \right ) L_2 \end{array} \right ], \end{equation}

and this shows that $F_z \equiv F^{\prime}_z$ does not hold only when $u_1 \equiv u_2$ does not hold. The second term of Eq. (42) means the degree of freedom of control input caused by increasing the number of the stance legs to two.

4.2. Approximate analytical solution of $T_1$

In this subsection, we derive an approximate analytical solution of $T_1$ for stabilizing the zero dynamics of the reaction wheel when $u_1 \equiv u_2$ . The equation of motion of the reaction wheel is

(43) \begin{equation} I_4{\ddot{\theta }}_4 = -u_2, \end{equation}

and the time integral for one step where $u_1 = u_2$ becomes

(44) \begin{equation} I_4 \left ( \dot{\theta }_{4} (T) - \dot{\theta }_{4} (0) \right ) = - \int _{0}^{T} u_1 \ \textrm{d}t, \end{equation}

where $T \;:\!=\; T_1 + T_2$ is the step period. Eq. (44) clearly shows that the necessary condition for the initial and terminal angular velocities to be equal becomes

(45) \begin{equation} J\;:\!=\; \int _{0}^{T} u_1 \ \textrm{d}t = 0. \end{equation}

An approximate analytical solution of $J$ can be derived by using the linearized reduced model [8] and is specified as

(46) \begin{equation} J= \frac{\alpha I_5 (1-\xi )}{T_1} - \frac{\alpha m g L_1}{12 T_1} \left ( 2 T_1^2 - 3 T_1 T_2 + T_2^2 \xi \right ). \end{equation}

By solving $J=0$ for $T_1$ , we obtain

(47) \begin{equation} T_1= \frac{3 T_2}{4} + \sqrt{ \frac{ mgL_1 T_2^2 \left ( 9 - 8 \xi \right ) + 96 I_5 \left ( 1 - \xi \right ) }{16mgL_1}}. \end{equation}

Table I. Physical and control parameters.

Figure 2. Simulation results of limit cycle walking where $A_\Delta =0$ and $b_\Delta = 0$ .

Figure 3. Snapshots of generated gait in Figure 2.

Figure 4. Simulation results of limit cycle walking where $A_\Delta = -0.0195577$ and $b_{\Delta } = 0.00060233$ .

Figure 5. Magnified views of Figure 4(d).

Figure 6. Phase-plane plot of generated motions.

4.3. Correction of resultant terminal state of reaction wheel

The state-space realization of Eq. (43) becomes

(48) \begin{equation} \frac{\textrm{d}}{\textrm{d}t} \left [ \begin{array}{c} \theta _4 \\ \dot{\theta }_4 \end{array} \right ] = \left [ \begin{array}{c@{\quad}c} 0 & 1 \\ 0 & 0 \end{array} \right ] \left [ \begin{array}{c} \theta _4 \\ \dot{\theta }_4 \end{array} \right ] + \left [ \begin{array}{c} 0 \\ -I_4^{-1} \end{array} \right ] u_2, \end{equation}

and we denote Eq. (48) as

(49) \begin{equation} \dot{\boldsymbol{x}}_r =\boldsymbol{A}_r\boldsymbol{x}_r +\boldsymbol{B}_r u_2. \end{equation}

In this paper, we consider adding a sinusoidal control input with an offset to $u_2$ as follows.

(50) \begin{equation} u_2 = u_1 + A_{\Delta } \sin \left ( \frac{2 \pi t}{T} \right ) + b_{\Delta } \end{equation}

When $A_{\Delta } = 0$ and $b_{\Delta } = 0$ , due to the gap between the linearized and nonlinear models, the resultant terminal angular velocity of the reaction wheel becomes off zero only a little, whereas the resultant terminal angular velocity is not immediately determinable. Let $\Delta \theta _4$ and $\Delta \dot{\theta }_4$ be the resultant terminal values of the angular position and angular velocity of the reaction wheel when starting from the origin on the phase plane. Then, the analytical solution of Eq. (49) with $\boldsymbol{x}_r (0) = \textbf{0}_{2 \times 1}$ where $A_{\Delta } = 0$ and $b_{\Delta } = 0$ or $u_2 = u_1$ should satisfy

(51) \begin{equation} \left [ \begin{array}{c} \Delta \theta _4 \\ \Delta \dot{\theta }_4 \end{array} \right ] = \int _{0}^{T}{{\textrm{e}}}^{\boldsymbol{A}_r \left ( T - t \right )}\boldsymbol{B}_r u_1 \ \textrm{d}t. \end{equation}

To control these values to zero, the sinusoidal control input should satisfy the following relationship.

(52) \begin{equation} \left [ \begin{array}{c} -\Delta \theta _4 \\ -\Delta \dot{\theta }_4 \end{array} \right ] = \int _{0}^{T}{{\textrm{e}}}^{\boldsymbol{A}_r \left ( T - t \right )}\boldsymbol{B}_r \left ( A_\Delta \sin \left ( \frac{2 \pi t}{T} \right ) + b_\Delta \right ) \textrm{d}t \end{equation}

By solving this for $A_\Delta$ and $b_\Delta$ , we obtain

(53) \begin{equation} A_\Delta = \frac{I_4 \pi \left ( 2 \Delta \theta _4 - T \Delta \dot{\theta }_4 \right )}{T^2}, \ \ b_\Delta = \frac{I_4 \Delta \dot{\theta }_4}{T}. \end{equation}

5. Analysis and discussions of gait characteristics based on numerical examples

Figure 2 shows the simulation results of limit cycle walking where $A_\Delta = 0$ and $b_{\Delta } = 0$ . The physical and control parameters were chosen as the values listed in Table I. Here, Figure 2(a) is the time evolution of $\theta _1 \equiv \theta _3$ , (b) that of $\dot{\theta }_1 \equiv \dot{\theta }_3$ , (c) that of $\theta _4$ , (d) that of $u_1$ and $u_2$ , (e) that of $F_z$ and $F^{\prime}_z$ , and (f) that of the ZMP. From Figs. 2(a) and (b), we can see that the stance-leg motion was perfectly controlled so that it starts from an upright stationary state and returns to the same state in one step, and $\theta _1 \equiv \theta _3$ and $\dot{\theta }_1 \equiv \dot{\theta }_3$ always hold during motion. From Figure 2(c), we can see that the angular position of the reaction wheel has returned to near zero, but this is not a natural result because $J=0$ is just a condition to satisfy $\dot{\theta }_4(0) = \dot{\theta }_4(T)$ . We can also see that the terminal value $\theta _4(T)$ moves away from zero with each step. Figure 2(d) shows that $u_1 \equiv u_2$ holds, and Figure 2(e) also shows that the resulting vertical ground reaction forces, $F_z$ and $F^{\prime}_z$ , are also equal to each other. Furthermore, Fig 2(f) shows that the final result of these is that the $X$ -position of the ZMP, $x + \frac{2 L_2 F^{\prime}_z}{F_z+F^{\prime}_z}$ , is always maintained at the midpoint between the rear and forefeet, $x$ and $x'$ .

Figure 3 shows the snapshots of the generated gait in Figure 2. It can be seen that the stance-leg motion is appropriately generated while maintaining the rear and fore vertical ground reaction forces indicated by the red arrows at the same value. From the last snapshot, however, we can see that the angular position of the reaction wheel has not returned to zero.

In the generated gait of Figure 2, the resultant terminal angular position and angular velocity of the reaction wheel were $\Delta \theta _4 = -0.400981$ [rad] and $\Delta \dot{\theta }_4 = 0.0719323$ [rad/s]. As previously mentioned, $\Delta \dot{\theta }_4$ is very close to zero because it comes from the gap between the linearized and nonlinear models. We then consider to correct these values to zero in the following. The target parameters of Eq. (50) were obtained as $A_\Delta = -0.0195577$ and $b_{\Delta } = 0.00060233$ [N $\cdot$ m]. Figure 4 shows the simulation results of limit cycle walking with terminal value correction. The system parameters were chosen as the values listed in Table I again. Since the generated stance-leg motion is the same as that of Figure 2, it is omitted here. Figure 4(a) shows that the angular position of the reaction wheel started from zero and successfully returned to zero according to the correction. Although the detailed result is omitted, the angular velocity also started from zero and returns to zero. Figure 4(b) shows that the time evolution of $u_1$ and $u_2$ is almost indistinguishable from that in Figure 2(b) because the values of $A_\Delta$ and $b_\Delta$ are very small compared to the ranges of $u_1$ and $u_2$ . This small difference in control input, however, clearly appears as a result of the different trajectories of $F_z$ and $F^{\prime}_z$ in Figure 4(c). Furthermore, Figure 4(d) shows that the final result of these is that the $X$ -position of the ZMP moves slightly back and forth around the midpoint between the rear and forefeet, $x$ and $x'$ , but it is also nearly impossible to distinguish. To clearly visualize that, magnified views of the generated ZMP trajectory are shown in Figure 5. We can see that a ZMP trajectory close to a sinusoidal wave with a small amplitude is generated.

Figure 6 shows the phase-plane plots of (a) the stance-leg motion, and (b) the reaction wheel motions with and without terminal value correction. Figure 6(a) shows that, since the stance-leg motion is preferentially controlled, it starts from the origin on the phase plane, jumps at the collision of stance-leg exchange, and returns to the origin again by DODC regardless of the reaction wheel motion. The dashed line in Figure 6(b) plots 20 s of the generated motion of the reaction wheel without terminal value correction, and we can see that it is gradually diverging. With terminal value correction, however, as shown by the solid line, the generated motion can be controlled to return to the origin in one step by adding a slight control input but almost without moving ZMP during motion. Figure 7 shows the snapshots of the generated gait in Figure 4. It can be seen that, from the last snapshot, the angular position of the reaction wheel has successfully returned to zero.

Figure 7. Snapshots of generated gait in Figure 4.

6. Conclusion and future work

In this paper, we introduced the planar rimless wheel type walker with parallel linkage legs and developed the control system for generating a walking motion that starts from an upright stationary state and returns to the same state in one step. The stance-leg motion was generated by control input such that the time integral of it for one step becomes zero. The reaction wheel motion as zero dynamics was also generated by the same control input, and the resultant terminal state was corrected by the additional sinusoidal control input. The validity of the mathematical results has been evaluated through numerical simulations.

In actual implementation, there is a possibility that the motions of the stance leg and reaction wheel cannot be returned to a perfectly stationary state due to various uncertainties. In such a case, a realistic countermeasure would be to introduce Phase III after Phase II and correct the resultant terminal state to the origin on the phase plane while maintaining the upright posture.

Now we are investigating control methods for sudden stops from high-speed walking and climbing stairs, and improving energy efficiency in realizing these. We are also considering extensions to bipedal robots that have the same parallel linkage legs.