Stable walking of 3D compass-like biped robot with underactuated ankles using discrete transverse linearization

Abstract

In this study, a walking manner of a spatial compass-like biped robot with underactuated ankles is presented. This biped robot has 3 degrees of freedom with only one actuator. The hip joint is equipped with a constraint mechanism to lock the angle when it retracts to a desired value. With the Poincare shooting method, a limit cycle of 3D walking gait on level ground is obtained. This limit cycle is unstable by checking the Jacobian of the Poincare map. We use this limit cycle as a reference trajectory and then introduce a method based on discrete transverse linearization to transform the reference trajectory tracking problem to a stabilization problem of a discrete periodic linear system. Thus, the classical discrete-time linear quadratic regulator is recalled to solve the stabilization problem. That is, a stable walking control law is obtained for the proposed 3D robot with a constraint mechanism.

Keywords

3D walking biped robot discrete transverse linearization DLQR

Introduction

Humanoid robotics have been investigated for decades, and lots of achievements have been obtained, such as these famous humanoid robots: ASIMO (Sakagami et al., 2002), QRIO (Nagasaka et al., 2004), and HUBO (Park et al., 2008). The walking gaits of these robots are generated by control strategies based on the zero moment point (ZMP) constraint. To use this kind of control strategies, the robots have large feet and move slowly. The main drawback of such a walking style is the large amounts of depletion of energy. For ASIMO, the specific mechanical cost of transport (C_mt = (positive mechanical energy)/(weight × distance traveled)) for walking is about 1.6, which is about 32 times of a typical human (0.05) (Collins et al., 2005). This issue is a significant problem that should be addressed further. In 1990, McGeer found that a biped walking machine can walk on a gently downhill slope without any input, and defined this kind of walking as passive dynamic walking (McGeer, 1990). In 1999, Kuo developed passive dynamic walking for a spatial biped robot (Kuo, 1999). Although the efficiency of passive dynamic walking is high, this walking gait can only be achieved when the robot walks on a downhill slope. It means that the application of passive walking gait is limited. However, it supports a feasible idea that robots can walk in a highly-efficient style.

As torque input is necessary for robot walking on level ground, researchers began to study quasi-passive walking of robots with underactuated ankles, and believed that the robot with underactuated ankles can accomplish a series of naturally high-efficient walking gaits with proper control. Grizzle et al. introduced the hybrid zero dynamic (HZD) method (Grizzle et al., 2001; Westervelt et al., 2003) to generate high efficient walking gaits for a planar robot with underactuated ankles. The time scaling control method (Chevallereau, 2003) was also an effective method to deal with this kind of problem. Later, these methods have been developed to be suitable for spatial biped robots (Chevallereau et al., 2009; Tlalolini et al., 2011) in achieving highly-efficient walking. Similar to the method of HZD, Aoyama and colleagues presented the method of passive dynamic autonomous control (PDAC) to control a spatial robot walking efficiently (Aoyama et al., 2009). Based on their methods, the efficiency of walking gaits is usually affected by the reference trajectories represented by Bezier polynomials.

In addition to the design of efficient reference walking gaits, some works have proposed to equip physical devices for the purpose of improving walking efficiency. For example, Kato and Ohtsuka commented that constraint mechanism helps to reduce the maximum input of the hip joint (Kato and Ohtsuka, 2009), but no stability analysis was given for biped walking in the presence of a constraint mechanism. Recently, Tang et al. investigated energy-efficient walking gaits of a compass-like biped robot with a constraint mechanism in the hip joint (Tang et al., 2012). However, such modification may lead to non-existence of a naturally stable walking limit cycle, therefore, extra control efforts are required to stabilize the unstable limit cycle. Event-based control (Chevallereau et al., 2009; Hu et al., 2011) is usually used to stabilize an unstable limit cycle, but the basin of attraction and the convergence rate is relatively conservative.Song and Zefran (2006) applied a kind of transverse linearization to stabilize the walking gaits of spatial biped robots, with the control law designed on the basis of discrete approximation of an equivalent linear system. As the stability of the new system obtained via discrete approximation is not equivalent to the stability of the original limit cycle, this method cannot guarantee stabilization of the system. Later, Manchester et al. introduced another kind of transverse linearization to control a planar compass-like biped robot with underactuated ankles (Manchester et al., 2011). However, this method has not been developed to control spatial robots with underactuated ankles.

Along the direction of passive walking and quasi-passive walking, we are struggling to develop an efficient walking manner for 3D robots with underactuated ankles. Based on the conclusion in (Kato and Ohtsuka, 2009), we believe that by adding a constrained mechanism on the hip to lock the hip joint when it retracts to a desired angle, we can further improve the walking efficiency. To test this idea, in this paper we consider a simple spatial compass-like biped robot with a constrained mechanism attached to the hip. With this setup, a highly-efficient walking limit cycle for the robot on the level ground is obtained based on the method of Poincare shooting. Unfortunately, directly checking the stability of the limit cycle with the Poincare map shows that this limit-cycle walking gait is not stable. We then propose to apply a discrete transverse linearization method for the stabilization purpose. That is, the entire limit cycle is intersected by a series of moving Poincare sections, and then a discrete linear periodic system is obtained by linearizing the system at the intersection points of a series of moving Poincare sections. By doing so, the stability of the limit cycle can be ensured when a corresponding discrete-time linear periodic system resulting from discrete transverse linearization is stable. The periodic feedback matrix gain of the linearized periodic system is designed by the method of DLQR to make the desired limit cycle walking gait stable.

The contribution of this paper is twofold. First, it is new that a constraint mechanism on the hip of a spatial robot is considered to have an efficient walking gait. We show in this paper that the specific mechanical cost of transport C_mt is approximately 0.0281 by this design, which is much smaller than those resulted from the ZMP based control for 3D fully-actuated robots. Second, a discrete transverse linearization method is applied to regulate the trajectories to the highly-efficient reference walking gait. From simulation, we notice that our method realize faster convergence and larger basin of attraction, compared to the methods of event-based control (Chevallereau et al., 2009; Hu et al., 2011).

Section II divides a complete walking step into four sub-phases and the dynamic model of each sub-phase is constructed respectively. In Section III, a limit cycle of the 3D walking gait is obtained, and its stability is checked. In Section IV, the discrete transverse linearization method is introduced, and the method of control synthesis is also presented to stabilize the entire 3D walking gait. Section V concludes the paper.

Hybrid dynamic model

A simplified model of a spatial compass-like biped robot is presented in Figure 1. This robot consists of two links: two legs terminated with “point feet” linked by the hip. The hip consists of a revolute joint with one DOF which is actuated. The width of the hip is nonzero. The stance leg is assumed to act as a passive pivot in the sagittal and frontal planes, with no degree of the z axis (i.e., no yaw motion). This model corresponds to the robot with small feet. As a result, the yaw rotation is inhibited by friction. As the feet are designed very small, we neglect the weights of them. When the stance foot touches the ground, it is flat on the ground with no ankle torque input. The hip joint is actuated, and a hip torque $u$ is applied between the stance leg and the swing leg. We suppose that a mechanism is equipped at the hip of the robot to keep the angle of the hip joint fixed when the joint retracts to a desired value. Note that this model is complicated enough to illustrate the walking gait features which do not occur in a planar robot. At the same time, it is also simple enough to keep the presentation of our control method transparent.

Figure 1.

A spatial compass-like biped robot.

In summary, the biped in the single support phase has only one degree of actuation and two degrees of underactuation. We define the angles $q_{1}$ [rad] and $q_{2}$ [rad] as the roll and pitch angles of the stance leg, respectively. That is, for a given global coordinate frame $x_{st} - y_{st} - z_{st}$ with its $z_{st}$ axis perpendicular to the ground, rotate it by $q_{0}$ [rad] about the $z_{st}$ axis at each step to get the local coordinate frame $x_{1} - y_{1} - z_{1}$ so that its $x_{1}$ axis is consistent to the direction of the stance foot. Then $q_{1}$ [rad] is defined to be the rotation angle of the stance leg around the $y_{1}$ axis in the local coordinate frame $x_{1} - y_{1} - z_{1}$ . Moreover, rotate the local coordinate frame $x_{1} - y_{1} - z_{1}$ by $q_{1}$ [rad] about the $y_{1}$ axis to get the local coordinate frame $x_{2} - y_{2} - z_{2}$ . Then $q_{2}$ [rad] is defined to be the rotation angle of the stance leg around the $x_{2}$ axis in the local coordinate frame $x_{2} - y_{2} - z_{2}$ .

The frontal plane in Figure 2a is define as the $x_{1} - z_{1}$ plane of the local coordinate frame 1, and the sagittal plane in Figure 2b is define as the $y_{2} - z_{2}$ plane of the local coordinate frame 2. The angle $q_{3}$ [rad] is defined as the joint angles of the hip relative to the stance leg about $x_{2}$ axis (i.e., the angle from the stance leg to the swing leg in the sagittal plane as shown in Figure 2b). In addition, the hip is always vertical to both the stance leg and the swing leg, resulting from only one DOF with actuation is configured in the hip joint. The positive angles are computed clockwise with respect to the vertical line.

Figure 2.

The three-dimensional compass-like biped robot shown in the frontal plane and sagittal plane.

Now we make some assumptions.

Each link is rigid and has mass.

All collisions are perfectly inelastic collisions (no bounce).

There is no slipping when the stance leg contacts the ground.

Transfer between the swing leg and the stance leg is instantaneous.

The gait is symmetric in steady state.

Walking takes place on a flat surface

Table 1.

Physical parameters for simulation.

Physical parameters	Hip mass	Leg mass	Hip width	Leg length
Variable	$m_{H}$	$m$	$h$	$l$
Value	$10 kg$	$5 kg$	$0.15 m$	$1 m$

In this model, if the angle of the hip $q_{3}$ retracts to the desired angle before the swing leg contacts the ground, there will be two collisions in a half of one gait step: the collision of hip and the collision of foot. When this robot walks, there are two sub-phases during a half of the whole walking cycle, which are shown in Figure 3.

Figure 3.

One half of one walking cycle.

Sub-phase A (hip joint unlocked): From the time instant just after the swing foot touches the ground to the time instant when the constraint mechanism in the hip joint locks. During this sub-phase, the swing leg swings from back to front. When the angle of the hip joint retracts to a desire value, the hip collision occurs.

Sub-phase B (hip joint locked): From the time instant when the constraint mechanism in the hip joint locks to the time instant just before the swing foot touches the ground. During this sub-phase, the hip joint is locked. When the swing leg touches the ground, the foot collision occurs.

The dynamic models for single support phase (i.e., sub-phase A and sub-phase B) and impacts (i.e., hip collision and foot collision) are derived here by assuming support on leg 1 (i.e., right leg in Figure 1). Define $q_{leg 1} = {[\begin{matrix} q_{1} & q_{2} & q_{3} \end{matrix}]}^{T}$ , and $q = q_{leg 1}$ when the robot is supported by leg 1. The models for support on leg 2 can be written in a similar way by using hip width of $- h$ in place of $h$ and $q = q_{leg 2}$ ( $q_{leg 2} = {[\begin{matrix} q_{1} & q_{2} & q_{3} \end{matrix}]}^{T}$ when the robot is supported by leg 2). The Euler-Lagrange equations for the robot walking can be obtained as follows (see, for example, Spong and Vidyasagar, 1989).

Sub-phase A: During sub-phase A, the hip joint is actuated. The dynamic model of the robot walking is shown as following

M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) = Bu

(1)

where $q = {[\begin{matrix} q_{1} & q_{2} & q_{3} \end{matrix}]}^{T}$ , $M (q)$ is the positive-definite $(3 \times 3)$ mass-inertia matrix, $C (q, \overset{\cdot}{q})$ is the $(3 \times 3)$ Coriolis matrix, $G (q)$ is the gravity term, $B = {(\partial q_{3} / \partial q)}^{T} = {[\begin{matrix} 0 & 0 & 1 \end{matrix}]}^{T}$ maps the hip joint torque to generalized forces, and $u$ is the input in the hip joint.

Hip collision: As the angle of hip joint retracts to a desired value, the hip collision occurs. During the collision, the biped’s configuration variables do not change, but the generalized velocities undergo a jump. Similar to formulating the impact model for the knee collision as developed in Yamakita and Asano (2001), the impulsive map for $q$ and $\overset{\cdot}{q}$ can be independently obtained at the time instant when the hip collision occurs. That is,

q^{+} = q^{-}

(2)

{\overset{\cdot}{q}}^{+} = {\overset{\cdot}{q}}^{-} - M {(q^{-})}^{- 1} J^{T} λ_{i}

(3)

where ‘−’ and ‘+’ represent the quantities before and after the hip collision, $J = \partial q_{3} / \partial q = [\begin{matrix} 0 & 0 & 1 \end{matrix}]$ , and $λ_{i} = {(JM {(q)}^{- 1} J^{T})}^{- 1} J {\overset{\cdot}{q}}^{-}$ is the torque impulse in the hip joint. So $J^{T} λ_{i}$ represents the impulsive force vector.

Sub-phase B: During sub-phase B, the hip joint is locked. The dynamic model of the robot walking is shown as follows:

M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) = Bu + J^{T} λ_{t}

(4)

where $λ_{t} = {(JM {(q)}^{- 1} J^{T})}^{- 1} JM {(q)}^{- 1} (- C (q, \overset{\cdot}{q}) \overset{\cdot}{q} - G (q))$ is the constraint torque. Because the hip joint is locked by the constraint mechanism, the input in the hip joint turns out to be $u = 0$ .

Foot collision: When the swing leg touches ground, the foot collision occurs. As with the hip collision, the angles of each joints do not change during the collision, and the generalized velocities undergo a jump. The position of the robot with respect to an inertial frame is required for calculating the angular velocities jump. The vector $q_{e}$ is defined by adding four variables $q_{e} = {[\begin{matrix} q^{T} & x_{st} & y_{st} & z_{st} & q_{0} \end{matrix}]}^{T}$ , where $x_{st}$ , $y_{st}$ and $z_{st}$ are the Cartesian coordinates of the stance leg. These additional variables are all constant during sub-phase A and B. The angular velocities of the joints during the foot collision can be calculated as follows (Chevallereau et al., 2009)

[\begin{matrix} {\overset{\cdot}{q}}_{e}^{+} \\ F_{sw} \end{matrix}] = {[\begin{matrix} D_{e} & - E_{sw}^{T} \\ E_{sw} & 0_{4 \times 4} \end{matrix}]}^{- 1} [\begin{matrix} D_{e} {\overset{\cdot}{q}}_{e}^{-} \\ 0_{4 \times 1} \end{matrix}]

(5)

where ‘−’ and ‘+’ represent the quantities before and after the foot collision, $F_{sw}$ is the impulsive reaction force at the contact point, $D_{e}$ is the extended mass-inertia matrix which is defined as the same as in Chevallereau et al. (2009), and $E_{sw} = \partial / \partial q_{e} {[\begin{matrix} x_{sw} & y_{sw} & z_{sw} & q_{0 sw} \end{matrix}]}^{T}$ is the Jacobian matrix for the position of the swing foot and its rotation angle about the $z$ -axis in the global frame.

Similarly to Chevallereau et al. (2009) and Grizzle et al. (2001), from equation (5), an impact map for foot collision can be defined as:

{\overset{\cdot}{q}}^{+} = Δ_{\overset{\cdot}{q}} (q^{-}, {\overset{\cdot}{q}}^{-})

(6)

After the foot collision, the biped’s legs exchange roles. Projecting the angle vector to the generalized coordinates for support leg 2 can be shown as follows:

q_{leg 2} = {Dq}_{leg 1}

(7)

where $D = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & - 1 \end{matrix}]$ . The angular velocities ${\overset{\cdot}{q}}_{leg 2}$ to the generalized coordinates for support leg 2 can also be calculated in the same way. With a similar procedure, the walking models for support on leg 2 can be obtained.

Limit cycle of walking gait

We know that the Poincare shooting method is an efficient way to find a limit cycle of passive running gait from (Hu et al., 2011, Yi et al., 2012). In this section, we obtain a limit cycle of walking gait in a similar way. Let $u_{h}$ be a torque value of the hip joint and keep constant during sub-phase A, and $θ_{Hd}$ be the desired value of the hip joint when the hip is locked. Denote by $z_{sw} (q_{leg 1})$ the height of the tip of the swing leg above the ground for the case when the stance leg at this moment is leg 1, and denote by $z_{sw} (q_{leg 2})$ the height of the tip of the swing leg above the ground for the case when the stance leg is leg 2. Then, the following two surfaces are introduced to define a discretized map to hunt a limit cycle. That is, $S_{1} = {(q_{leg 1}, {\overset{\cdot}{q}}_{leg 1}) | z_{sw} (q_{leg 1}) = 0}$ and $S_{2} = {(q_{leg 2}, {\overset{\cdot}{q}}_{leg 2}) | z_{sw} (q_{leg 2}) = 0}$ , respectively, represent the sets of states, for which the swing foot touches the ground and a foot collision occurs.

With the above definition of the state sets $S_{1} = {(q_{leg 1}, {\overset{\cdot}{q}}_{leg 1}) | z_{sw} (q_{leg 1}) = 0}$ and $S_{2} = {(q_{leg 2}, {\overset{\cdot}{q}}_{leg 2}) | z_{sw} (q_{leg 2}) = 0}$ , then we are able to present the discretized map for the bipedal walking motion as

x_{d} (2) = f_{d} (x_{d} (1), u_{h}, θ_{Hd}),

where $x_{d} (2)$ is a state vector in $S_{2}$ , $x_{d} (1)$ is a state vector in $S_{1}$ , and $f_{d} : S_{1} \to S_{2}$ is a map from $S_{1}$ to $S_{2}$ capturing the evolution of the states. This map can be obtained by calculating the dynamic system of robot walking. As the walking gait is symmetric in steady state, the following equations must be satisfied when the states are in a limit cycle of walking gait:

x_{d}^{*} (2) = [\begin{matrix} - 1 & 0_{1 \times 2} & 0 & 0_{1 \times 2} \\ 0_{2 \times 1} & I_{2 \times 2} & 0_{2 \times 1} & 0_{2 \times 2} \\ 0 & 0_{1 \times 2} & - 1 & 0_{1 \times 2} \\ 0_{2 \times 1} & 0_{2 \times 2} & 0_{2 \times 1} & I_{2 \times 2} \end{matrix}] x_{d}^{*} (1) = f_{d} (x_{d}^{*} (1), u_{h}, θ_{Hd}),

(8)

where $I_{2 \times 2}$ is an unit matrix, and $x_{d}^{*} (1)$ is the state in the limit cycle.

Define a nonlinear function as

\begin{matrix} Nc (x_{d} (1), u_{h}, θ_{Hd}) \\ = ‖ f_{d} (x_{d} (1), u_{h}, θ_{Hd}) - [\begin{matrix} - 1 & 0_{1 \times 2} & 0 & 0_{1 \times 2} \\ 0_{2 \times 1} & I_{2 \times 2} & 0_{2 \times 1} & 0_{2 \times 2} \\ 0 & 0_{1 \times 2} & - 1 & 0_{1 \times 2} \\ 0_{2 \times 1} & 0_{2 \times 2} & 0_{2 \times 1} & I_{2 \times 2} \end{matrix}] x_{d} (1) ‖ . \end{matrix}

The state $x_{d}^{*} (1)$ and the parameters $u_{h}^{*}$ and $θ_{Hd}^{*}$ in the limit cycle can be calculated by solving the equation $Nc (x_{d}^{*} (1), u_{h}^{*}, θ_{Hd}^{*}) = 0$ . However, this nonlinear equation is difficult to solve. In this paper, the limit cycle is hunted with a constraint optimization method, and the following constraint optimization problem is defined:

\begin{matrix} minimize : & Nc (x_{d} (1), u_{h}, θ_{Hd}), \\ subject to : & Nc (x_{d} (1), u_{h}, θ_{Hd}) = 0, \\ {\overset{\cdot}{q}}_{2} > 0, \\ F_{z} > 0, \\ | \frac{\sqrt{F_{x}^{2} + F_{y}^{2}}}{F_{z}} | < μ, \end{matrix}

where $F_{x}$ , $F_{x}$ and $F_{x}$ are the $x$ , $y$ and $z$ direction components of ground reaction force, and $μ$ is the coefficient of friction. And, the inequality conditions can be satisfied for walking with a small input.

Using the FMINCON funciton in Matlab, a feasible limit cycle walking gait is obtained. Then, enlarge $| θ_{Hd} |$ along one direction, and solve the optimization problem until there is no solution. Next, fix $| θ_{Hd} |$ , and solve the optimization problem by reducing $| u_{h} |$ . Finally, a target limit cycle walking gait with $x_{d}^{*} (1)$ , $u_{h}^{*}$ and $θ_{Hd}^{*}$ can be obtained. For the biped robot described in the paper, Figure 4 shows an optimal limit-cycle walking gait.

Figure 4.

The 3D graph of an optimal walking motion for the biped robot.

The specific mechanical cost of transport $C_{mt}$ introduced in Collins et al. (2005) is a kind of coefficient to check the efficiency of the walking gait, where $C_{mt}$ is defined below.

C_{mt} = \frac{\int_{0}^{T} E (t) dt}{(m_{H} + 2 m) gL},

where $T$ is the time period of the limit-cycle walking gait, and $L$ is the step length, and $E (t)$ is the positive power of joint actuation, which satisfies

{\begin{matrix} E (t) = u_{h} {\overset{\cdot}{q}}_{3} & if u_{h} {\overset{\cdot}{q}}_{3} > 0, \\ E (t) = 0 & otherwise . \end{matrix}

For the walking gait in Figure 4, the robot walks with the speed about 0.29m/s and $C_{mt} = 0.0281$ . This walking gait is evidently very efficient. From the reference (Aoyama et al., 2009), $C_{mt}$ for 3D robot is summarized in Table 2:

Table 2.

$C_{mt}$ of 3D robots for different methods in the reference.

Method	$C_{mt}$
ZMP based control (Asimo)	1.6
LIMP based control (Gorilla robot)	0.57
PDAC based control (Gorilla robot)	0.15
(Cornell powered biped)	0.055

Stability of the limit cycle is the key factor for the biped robot walking. To check stability, the Poincare map is used. Denote by $S_{b}$ the Poincare surface, and define a map $f_{b} : S_{b} \to S_{b}$ . This map can be obtained with the method similar as $f_{d}$ . The Poincare map is

x_{b}^{*} = {Sf}_{b} (x_{b}^{*}) : = P_{b} (x_{b}^{*}),

(9)

where $S = diag {- 1, 1, - 1, 1, 1}$ , and $x_{b}^{*}$ is the fixed point in the Poincare surface. The eigenvalues of the Jacobian of the Poincare map $D_{x} P (x_{b}^{*})$ are then used to check the stability of the limit cycle. In this study, we select the Poincare surface $S_{b} = {(q, \overset{\cdot}{q}) | q_{2} = 0}$ , and calculate the Jacobian of the Poincare map $D_{x} P (x_{b}^{*})$ using the numerical method. Consider the limit cycle of the walking gait in Figure 4, the fixed point in the selected Poincare surface for supporting on leg 1 is

x_{b}^{*} = [\begin{matrix} 0.0472 \\ - 0.3874 \\ - 0.0971 \\ 0.2672 \\ - 0.2983 \end{matrix}] .

And the Jacobian of the Poincare map $D_{x} P (x_{b}^{*})$ is calculated as:

\begin{matrix} D_{x} P (x_{b}^{*}) \\ = [\begin{matrix} - 5.7290 & 0.3832 & - 2.0393 & - 0.6256 & 0.0664 \\ 4.0700 & - 0.3055 & 0.8756 & 0.9286 & - 0.1112 \\ - 15.8581 & 1.0452 & - 6.2202 & - 1.3766 & 0.1362 \\ - 6.8667 & 0.5023 & - 1.5129 & - 1.4549 & 0.1732 \\ - 132.8151 & 9.8583 & - 29.1090 & - 29.2542 & 3.4926 \end{matrix}] . \end{matrix}

Let $λ_{i} (i = 1, 2 \dots 5)$ be the eigenvalues of $D_{x} P (x_{b}^{*})$ . The maximal absolution value of the eigenvalues $max (| λ_{i} |) = 11.3928$ is outside the unit circle, we know that the limit cycle is unstable. The unstable trajectories are shown in Figures 5 –7.

Figure 5.

The nominal limit cycle is unstable. If the initial conditions are not in the limit cycles, the trajectory in $q_{1}$ diverges.

Figure 6.

The nominal limit cycle is unstable. If the initial conditions are not in the limit cycles, the trajectory of $q_{2}$ diverges.

Figure 7.

The nominal limit cycle is unstable. If the initial conditions are not in the limit cycles, the trajectory of $q_{3}$ diverges.

Figures 5 –7 show the phase-plane plots for $q_{1}$ , $q_{2}$ and $q_{3}$ with the initial state closed to the limit cycle. After several steps, the trajectories diverge from the limit cycle, and the robot will fall down. As unstable walking gaits are useless for robots, in order to stabilize the limit cycle with a good dynamic performance, a stabilization control strategy of discrete transverse linearization is presented in the next section.

Discrete transverse linearization

In this section, a control method is proposed to stabilize the limit cycle without changing the structure of the robot. Considering that the event-based control has the drawback of limitation of the number of input regulation, in this paper, the desired limit cycle is intersected by a series of moving Poincare sections, and the number of input regulation can be increased. Hence, based on a proper control strategy, the convergence rate can be accelerated, and the basin of attraction can be enlarged.

General method

Consider a nonlinear system

\overset{\cdot}{x} = f (x, β),

(10)

where $β$ is an adjustable parameter vector.

Let $x^{*} (t)$ be a periodic solution with $β = β^{*}$ . If the periodic solution is unstable, the discrete transverse linearization can be used to stabilize it. The method is presented as follows:

As shown in Figure 8, construct a series moving Poincare sections $S_{i} (i \in [\begin{matrix} 1 & N \end{matrix}])$ along the periodic solution $x^{*} (t)$ , $t \in [\begin{matrix} 0 & T \end{matrix}]$ , where $T$ is the period. Then, for each two adjacent moving Poincare sections, we can define a map $P_{k} : S_{k} \to S_{k + 1}$ ( $k = k + N$ ). Hence, a series of periodic map $x_{⊥} (k + 1) = P_{k} (x_{⊥} (k), β)$ with period $N$ can be constructed, i.e., $P_{k + N} (\cdot) = P_{k} (\cdot)$ , where $x_{⊥} (i)$ is the state in each moving Poincare section $S_{i}$ .

Figure 8.

The limit cycle can be intersected by a series of moving Poincare sections $S_{i} (i \in [\begin{matrix} 1 & N \end{matrix}])$ , and a series periodic map $x_{⊥} (k + 1) = P_{k} (x_{⊥} (k), β)$ with period $N$ can be constructed.

For the periodic solution, it is clear that:

x_{⊥}^{*} (k + 1) = P_{k} (x_{⊥}^{*} (k), β^{*}),

(11)

where $x_{⊥}^{*} (i)$ is the state in the periodic solution.

As the map $x_{⊥} (k + 1) = P_{k} (x_{⊥} (k), β)$ with period $N$ represent the original nonlinear system. A map can be constructed by expanding the state as following:

y (k) = [\begin{matrix} x_{⊥} (k) \\ x_{⊥} (k + 1) \\ x_{⊥} (k + 2) \\ ⋮ \\ x_{⊥} (k + N - 1) \end{matrix}]

Then, a discrete nonlinear system can be constructed as:

\begin{matrix} y (k + 1) & = [\begin{matrix} x_{⊥} (k + 1) \\ x_{⊥} (k + 2) \\ x_{⊥} (k + 3) \\ ⋮ \\ x_{⊥} (k + N) \end{matrix}] = [\begin{matrix} P_{k} (x_{⊥} (k), β) \\ P_{k + 1} (x_{⊥} (k + 1), β) \\ P_{k + 2} (x_{⊥} (k + 2), β) \\ ⋮ \\ P_{k + N - 1} (x_{⊥} (k + N - 1), β) \end{matrix}] \\ = η (y (k), β) \end{matrix}

Considering the points in the limit cycle $x_{⊥}^{*} (k)$ , $x_{⊥}^{*} (k + 1)$ , $x_{⊥}^{*} (k + 2)$ , … $x_{⊥}^{*} (k + N - 1)$ with $β = β^{*}$ , $y^{*} (k)$ is the fixed point for the discrete nonlinear system with $β = β^{*}$ . The stability of $y^{*} (k)$ indicates the stability of the limit cycle. Hence, it is clear that the stability of $x_{⊥}^{*} (i)$ represent the stability of the limit cycle.

Define $δ x_{⊥} (k) = x_{⊥} (k) - x_{⊥}^{*} (k)$ and $δ β (k) = β (k) - β^{*}$ , and linearize equations (11) around the periodic solution in each moving Poincare section with the similar method as the event-based control method (Chevallereau et al., 2009). Then, a discrete periodic linear system can be obtained.

δ x_{⊥} (k + 1) = G (k) δ x_{⊥} (k) + H (k) δ β (k)

(12)

where $G (k) = [\begin{matrix} G {(k)}_{1} & \dots & G {(k)}_{n} \end{matrix}] \in R^{n \times n}$ and $H (k) = [\begin{matrix} H {(k)}_{1} & \dots & H {(k)}_{m} \end{matrix}] \in R^{n \times m}$ are periodic with period $N$ , i.e., $G (k + N) = G (k)$ and $H (k + N) = H (k)$ , and $n$ is the dimension of $x_{⊥}^{*} (k)$ , and $m$ is the dimension of $β^{*}$ . The $i$ th element of $G {(k)}_{i}$ and $H {(k)}_{i}$ can be written as

G {(k)}_{i} = \frac{P_{k} (x_{⊥}^{*} (k) + Δ x_{⊥ i} (k), β^{*}) - P_{k} (x_{⊥}^{*} (k) - Δ x_{⊥ i} (k), β^{*})}{2 δ x_{⊥ i} (k)},

H {(k)}_{i} = \frac{P_{k} (x_{⊥}^{*} (k), β^{*} + Δ β_{i} (k)) - P_{k} (x_{⊥}^{*} (k), β^{*} - Δ β_{i} (k))}{2 δ β_{i} (k)},

where $δ x_{⊥ i} (k)$ and $δ β_{i} (k)$ are the $i$ th element of $δ x_{⊥} (k)$ and $δ β (k)$ , respectively. $Δ x_{⊥ i} (k)$ and $Δ β_{i} (k)$ are the vectors with the dimension as $x_{⊥}^{*} (k)$ and $β^{*}$ , respectively. The $i$ th element of $Δ x_{⊥ i} (k)$ and $Δ β_{i} (k)$ are $δ x_{⊥ i} (k)$ and $δ β_{i} (k)$ , respectively, while others are equal to zero.

Form the theorem of nonlinear system, we know that the stability of the Linearize system represent the stability of the original nonlinear system. Hence, If the discrete periodic linear system can be stabilized with feedback control $δ β (k) = - K (k) δ x_{⊥} (k)$ , the periodic solution of dynamic system (10) will be stabilized.

In order to stabilize the discrete periodic linear system, we resort to the method of DLQR, and the feedback gains $K (k)$ can be designed as follows (Kwakernaak and Sivan, 1972):

First, define the cost function as:

J = \sum_{i = 1}^{\infty} (δ x_{⊥}^{T} Q δ x_{⊥} + δ β^{T} R δ β),

(13)

where $Q$ is a semi-positive definite matrix, and $R$ is a positive definite matrix.

Then, the discrete Riccati equation can be written as

\begin{matrix} P (k) = & {[G (k) - H (k) K (k)]}^{T} P (k + 1) [G (k) - H (k) K (k)] \\ + Q + K^{T} (k) RK (k) . \end{matrix}

(14)

And the feedback gait can be calculated as

K (k) = {[R + H^{T} (k) P (k + 1) H (k)]}^{- 1} H^{T} (k) P (k + 1) G (k) .

(15)

Let $k + 1$ be an enough large positive integer, and set $P (k + 1) = 0$ , and iterate the equations (14) and (15). Finally, $P (k)$ will converge to a periodic matrices, and a periodic series of $K (k)$ can be obtained to stabilize the periodic solution.

Simulation result

In order to stabilize the limit-cycle walking gait, the moving Poincare sections $S_{k}$ are constructed firstly. For the limit cycle in Figure 4, $q_{2}^{*}$ is monotonous during the sub-phase A with support leg 1 and leg 2. Also, it is fact that $min (q_{2}^{*}) < - 15 \circ$ and $14 \circ < max (q_{2}^{*}) < 15 \circ$ during the sub-phase A. As the walking gait is symmetric in the limit cycle, $N$ must be a positive even integer. If $N / 2$ is an even integer, the moving Poincare sections as $S_{k} = {(q, \overset{\cdot}{q}) | q_{2} = \frac{15 π / 360}{N / 4} (k - 1 - \frac{N}{4}), (1 \leq k \leq \frac{N}{2})}$ and $S_{k} = {(q, \overset{\cdot}{q}) | q_{2} = \frac{15 π / 360}{N / 4} (k - 1 - \frac{3 N}{4}), (\frac{N + 2}{2} \leq k \leq N)}$ . On the other hand, if $N / 2$ is an odd integer $S_{k} = {(q, \overset{\cdot}{q}) | q_{2} = \frac{14 π / 360}{(N - 2) / 4} (k - 1 - \frac{N - 2}{4}), (1 \leq k \leq \frac{N}{2})}$ and $S_{k} = {(q, \overset{\cdot}{q}) | q_{2} = \frac{14 π / 360}{(N - 2) / 4} (k - 1 - \frac{3 N - 2}{4}), (\frac{N + 2}{2} \leq k \leq N)}$ . In each moving Poincare section $S_{k}$ , $q_{2}$ is constant. Hence, $x_{⊥}^{*} (k)$ can be defined as $x_{⊥}^{*} (k) = {[\begin{matrix} q_{1}^{*} & q_{3}^{*} & {\overset{\cdot}{q}}_{1}^{*} & {\overset{\cdot}{q}}_{2}^{*} & {\overset{\cdot}{q}}_{3}^{*} \end{matrix}]}^{T}$ . In the limit cycle of walking gait, the input $u = u^{*}$ is constant. Hence, let $β^{*} = u^{*}$ . Then, the discrete nonlinear system (11) can be obtained by calculating the dynamic system of robot walking with the numerical method. Linearizing the system (11), the discrete periodic linear system (12) can be calculated with the numerical method in last subsection with $G (k) \in R^{5 \times 5}$ and $H (k) \in R^{5 \times 1}$ . And with the DLQR method (13)–(15), the limit cycle of the walking gait can be stabilized. The schematic diagram of control flow is shown in Figure 9.

Figure 9.

The schematic diagram of control flow.

And, the stable trajectories are shown in Figures 10 –12, for $N = 8$ , $Q = I^{5 \times 5}$ and $R = 1$ .

Figure 10.

The trajectory of q₁ converges to the nominal limit cycle of the walking gait from an initial state perturbed away from the fixed point.

Figure 11.

The trajectory of q₂ converges to the nominal limit cycle of the walking gait from an initial state perturbed away from the fixed point.

Figure 12.

The trajectory of q₃ converges to the nominal limit cycle of the walking gait from an initial state perturbed away from the fixed point.

A comparison to the event-based control

In order to demonstrate the advantage of the method of discrete transverse linearization, a comparison between the method of discrete transverse linearization and the event-based control is made. Next, we briefly introduce the event-based control. Consider the following Poincare map for the limit cycle with a constant control input explicitly in the map,

x_{b}^{*} = P_{b} (x_{b}^{*}, β^{*}) .

(16)

Similar to the linearization method in deriving (12), a discrete time-invariant linear control system is obtained, that is,

δ x_{b} (k + 1) = G_{b} δ x_{b} (k) + H_{b} δ β (k) .

(17)

The event-based control law takes the form of $δ β (k) = - K_{b} δ x_{b} (k)$ to make the eigenvalues of $(G_{b} - H_{b} K_{b})$ strictly inside the unit circle. Thus, the fixed point $x_{b}^{*}$ is stabilized. In the above event-based control law, the gain matrix $K_{b}$ can be calculated via the DLQR approach to minimize $J = \sum_{i = 1}^{\infty} (δ x_{b}^{T} Q δ x_{b} + δ β^{T} R δ β)$ . To summary, the event-based control is applied as follows.

{\begin{matrix} δ β (k) = - K_{b} (x_{b} (k) - x_{b}^{*}) & for supporting on leg 1, \\ δ β (k) = - K_{b} ({Sx}_{b} (k) - x_{b}^{*}) & for supporting on leg 2 2 . \end{matrix}

(18)

For $x_{d}^{*} (1) \in S_{1}$ and $x_{d}^{*} (2) \in S_{2}$ which are the states in the limit cycle, considering the robot supported by leg 1, the norm of error $E_{N} (i)$ in the $i$ th step can be defined as follows:

E_{N} (i) = {\begin{matrix} ‖ x_{d} (i) - x_{d}^{*} (1) ‖ & for i is a positive odd integer, \\ ‖ x_{d} (i) - x_{d}^{*} (2) ‖ & for i is a positive even integer . \end{matrix}

The trajectory of $E_{N} (i)$ along the steps for the method of event-based control and the discrete transverse linearization are shown in Figure 13, and the basins of attraction for two methods are compared in Figure 14.

Figure 13.

The comparison of the norm of error $E_{N} (i)$ varying with the steps for two methods.

Figure 14.

Comparison of the basin of attraction for two methods: the black area is the basin of attraction for event-based control, and the gray area is the basin of attraction for discrete transverse linearization.

From Figure 13 and Figure 14, it is clear that the method of discrete transverse linearization has a faster convergence rate and a larger basin of attraction compared with event-based control.

A number of grids can be divided averagely within the range of $- 0.01 [rad] \leq Δ q_{3} \leq 0.01 [rad]$ and $- 0.02 [rad] \leq Δ {\overset{\cdot}{q}}_{3} \leq 0.02 [rad]$ , and the number of the grids represents the size of basin of attraction. As $N$ must be a positive even integer, define $N = 2 n$ . Increasing $n$ from 1 to 6, the size of basin of attraction according to the number of $n$ is shown in Figure 15, and the number of steps converging to $E_{N} (i) \leq 2 \times 10^{- 6}$ with the same initial condition according to the number of $n$ is shown in Figure 16.

Figure 15.

The number of the grid in the basin of attraction according to the number of n.

Figure 16.

The number of steps converging to $E_{N} (i) \leq 2 \times 10^{- 6}$ with the same initial condition according to the number of n.

From Figure 15, it can be seen that the basin of attraction increases obviously when $n$ increases from 1 to 3, but after $n = 4$ , the basin of attraction does not change obviously. From Figure 16, the convergence rate accelerates with $n$ from 1 to 2, and then, the convergence rate does not change obviously. Hence, the basin of attraction and the convergence rate for the discrete transverse linearization are all better than the method of event-based control. In this paper, we design the control strategy with $n = 4$ ( $N = 8$ ), as the basin of attraction and the convergence rate are all nearly best for $n$ from 1 to 6.

Conclusion

The walking problem of a spatial compass-like biped robot with underactuated ankles is studied. This biped robot has 3DOF with only one actuator in the hip. In order to keep the posture of the robot when the swing leg touches the ground, a constraint mechanism was equipped in the hip joint. Moreover, this mechanism is also benefit for reducing the input torque. With the method of Poincare shooting, an open loop limit cycle of 3D walking gait on level ground is obtained, and this waking gait is very efficient with the specific mechanical cost of transport $C_{mt} = 0.0281$ . In order to stabilize the limit cycle, a method of discrete transverse linearization was presented. As a result, the problem of limit cycle tracking was equivalent to the stabilization problem of a discrete periodic linear system. Based on the DLQR method, the limit cycle of 3D walking gait was finally stabilized. Compared to the general method of event-based control, our method presented in this paper has a larger basin of attraction and a faster convergence rate.

Footnotes

Conflict of interest

None declared.

Funding

The work is supported by National Natural Science Foundation of China under grant number 61175106.

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