Walking Algorithm of Humanoid Robot on Uneven Terrain with Terrain Estimation

Abstract

Humanoid robots are expected to achieve stable walking on uneven terrains. In this paper, a control algorithm for humanoid robots walking on previously unknown terrains with terrain estimation is proposed, which requires only minimum modification to the original walking gait. The swing foot trajectory is redesigned to ensure that the foot lands at the desired horizontal positions under various terrain height. A compliant terrain adaptation method is applied to the landing foot to achieve a firm contact with the ground. Then a terrain estimation method that takes into account the deformations of the linkages is applied, providing the target for the following correction and adjustment. The algorithm was validated through walking experiments on uneven terrains with the full-size humanoid robot Kong.

Keywords

Humanoid Robot Uneven Terrain Walking Control Terrain Estimation

1. Introduction

Achieving movements in human environments is one of the most important goals in humanoid robot research. However, walking on uneven terrains, especially on previously unknown terrains, is highly challenging. Therefore related works are relatively less fruitful while abundant researches have been made on walking speed and other aspects for walking on even terrains. The DAPAR Robotics Challenge (DRC) for disaster response scenarios showed that the capability to traverse different types of terrains is crucial for robots to assist humans in harsh environments [1], and yet such capability is still quite limited. The capability of adapting to uneven terrains is then raised as a key factor in the development of practical robots. Although vision and laser sensors can be assists for robots to perceive the environments, the problems of inaccuracy, hardware and computational cost limit the effectiveness. Hence, it is still necessary to study biped walking on uneven terrains without complete and reliable information of the environment.

Previous works on biped walking on uneven terrain can be divided into two major categories according to their different control methods: the dynamics-based methods and the model-based methods [2]. The dynamics-based methods, includes the passive walking [3], the Central Pattern Generator (CPG) method [4], the Virtual Model Control (VMC) method [5], etc. These methods achieve walking on uneven terrains through the robustness of their systems. And in some studies the model of the terrain was already given [6]. The stability analyses of these methods are complex, and usually rely on precise dynamics models of both the robots themselves and their impacts with the ground. Therefore, the implementation results of these methods can be considerably uncertain.

The majority of the model-based methods are variants of the Zero Momentum Point (ZMP) method. These methods achieve walking on uneven terrains by gait adjustments, or by using specially designed foot mechanisms [7, 8], which will not be discussed here in consideration of universality. The former can be further classified by whether modeling the uneven terrains with available sensor information or not. The methods without modeling the terrain focus on the negative effects caused by the non-ideality of the terrains. Hence they actually handle the uneven terrains along with all other non-ideality factors. Walking is stabilized by body acceleration [9, 10, 11], body rotation [12], or foot landing point adjustment in these methods [13, 14, 15]. On the contrary, the methods involving terrain modeling can make adjustments according to the conditions of the uneven terrains. Besides those methods obtaining terrain information from vision [16] or laser [10, 17, 18] as mentioned above, previous researches usually acquired terrain information from both the force-torque information during the foot landing period and the attitude information during the following single support phase [19, 20]. Accordingly, walking was stabilized by both instant adjustments in supporting foot posture and later adjustments in gait such as landing time. A “global terrain” was usually introduced in these works which assumed that there is a general trend in the variation of the terrain. This assumption actually reduced the difficulty of terrain modeling since previous terrain information can be used to reduce the uncertainty of terrain at current landing point. Impedance control is widely applied to achieve compliant foot landing in all these model-based methods.

The methods without terrain modeling lower the complexity of the control systems. However, as these methods are not specific to the conditions of the terrains, the potential to maintain the original gaits is actually lost. Meanwhile, variable walking parameters like step length and walking cycle period are modified, which might leave less margin for resisting other disturbances. On the contrary, the methods involving terrain modeling require less change to the trajectories of the Center of Mass (COM) and save the variable parameters for both faster walking and resisting disturbances. However, acquiring the terrain information is not easy yet, especially when demanded to be accomplished right after the landing of the foot. Some researchers thought it is difficult or even impossible to model the terrain accurately with only the information gathered during the short landing period [19].

We found that there are two major difficulties in building an online terrain model. First, rapid adaptation to the uneven terrain without hard impact is a challenge. Second, the elastic deformations of the legs that reduce the reliability of the kinematics should be taken into consideration. Hence, a new adaptation method based on the classic impedance control is proposed, which eliminates the coupling between the roll/pitch rotation and the vertical displacement by the feedforward strategy. This method prevents the repeated impact between the landing foot and the terrain, and also shortens the adaptation period. Meanwhile, a new terrain estimation method is designed, which calculates the decline of the landing foot through the rotation about the Center of Pressure (COP) rather than the normal forward kinematics using just the joint angles. The terrain estimation is accurate enough that only the height of the COM needs to be adjusted, so the original offline gait generated by the previewed control method [21] is highly preserved. The swing foot trajectory is also redesigned to cope with the uncertainty of the terrain.

The remainder of the paper is organized as follows. Section 2 gives an overview of all the proposed methods and a framework of the control system. Section 3 describes a redesigned swing foot trajectory. Section 4 introduces the terrain adaptation method in detail. In section 5, the terrain estimation method is described, along with the corresponding correction and adjustment strategies. In section 6, the experimental results are given and discussed. Finally, section 7 concludes the paper.

2. Overview of Methods

In this section, the assumptions of the problem and the framework of the control system are given.

2.1. Assumptions

In this paper, the robot is desired to walk on uneven terrains in the same manner as it does on even floor. There are many kinds of uneven surface conditions for robots. The following assumptions are made to restrict and simplify the problem:

The trend of the terrain is stochastic.

The height range of the terrain within the projection of the landing foot, represented by $[H_{l o w e r}, H_{u p p e r}]$ , should fall within a certain range around the height of the supporting foot H₀, represented by $[H_{0} - Δ H_{m i n}, H_{0} + Δ H_{m a x}]$ , as shown in Figure 1. $Δ H_{m i n}$ and $Δ H_{m a x}$ are given parameters limited by the max velocity and acceleration of the swing foot, which are determined by the hardware performance.

The support region is big enough to provide enough ZMP margin after the landing foot made a firm contact with the ground.

The stiffness of the terrain is consistent and is hard enough to avoid visible deformations.

Figure 1.

Scheme of the uneven terrain

2.2. Framework of control system

A state machine for the control system is designed, as shown in Figure 2. Comparing to the traditional state machine for biped walking, there are two differences:

Touching phase is introduced for the adaptation process of the landing foot.

The states for the left and the right foot are separated without the double support phase connecting them, as it is possible that one foot enters the flight phase when the other hasn't finished the touching phase.

Figure 2.

State machine of the control system

The framework of the control system is shown in Figure 3. In this control process, the preview control method is used to generate the offline COM trajectory with the gait parameters, like the step length and the scheduled landing time (which is the deadline for the actual landing). To realize walking on an uneven terrain, the following methods should be applied at different stages:

Redesigned swing foot trajectory. Due to the height variation of the terrain, the landing moment can be earlier or later than expected [22]. To ensure the swing foot touches the ground at desired horizontal position before the scheduled landing time, the swing foot trajectory should be redesigned.

Compliant adaptation of the landing foot to the terrain. The landing foot should actively adapt to the uneven terrain using the force-torque sensor information after it touches the ground.

Correction and adjustment based on terrain estimation. The pose of the landing foot calculated through the forward kinematics is not reliable because of the non-ideality factors such as the deformation of the linkages and the inclination of the body. And the height of the COM should change according to the variation of the terrain. Therefore corresponding corrections and adjustments are needed.

Figure 3.

Diagram of the overall control framework

3. Redesigned swing foot trajectory

As mentioned above, the swing foot should have touched the ground at the desired horizontal position before the scheduled landing time, as long as the height of the terrain falls into the allowed range. Therefore the horizontal components of the trajectory should reach the target value before the vertical component does. And the end of the planned trajectory should reach the lower limit of the terrain to guarantee a contact. Additional requirements for the swing foot trajectory are given as follows:

The continuity of the velocity should be guaranteed.

The maximum vertical decline velocity should be limited to avoid hard impacts.

The swing foot should not stay near the ground for too long in order to avoid repeated impacts with the ground.

Components of the trajectory other than the vertical displacement are all set to be sinusoids that reach the target values before the scheduled landing time and then hold the target values, for example, the displacement in the front-rear direction $x (t)$ . Given the initial value x₀ and the target value x₁, along with the start time of the step set to be zero for convenience

x (t) = w_{0} (t) x_{0} + [1 - w_{0} (t)] x_{1},

(1)

where

w_{0} (t) = {\begin{matrix} 0.5 + 0.5 \cos \frac{π t}{α T_{s t e p}}, & t \leq α T_{s t e p} \\ 0, & t > α T_{s t e p} \end{matrix}

(2)

where $T_{s t e p}$ is the scheduled landing time of the step and α is a factor that is set to 0.75 in this research. Other components of the trajectory, including the displacement in the left-right direction and all the rotations, can be calculated by replacing x₀ and x₁ with corresponding values.

We set the vertical displacement component of the trajectory as a piecewise function composed of four parabolas, as shown in Figure 4. The target value z₁ should be the minimum terrain height allowed

z_{1} = z_{s u p} - Δ H_{m i n},

(3)

where $z_{s u p}$ is the height of the supporting foot.

Given the initial value z₀, the target value z₁, the height of the trajectory $h_{t r a j}$ , the maximum decline velocity $V_{l o w e r}$ and the residence time $t_{n e a r}$ within the distance $h_{n e a r}$ from the ground, we have

z (t) = {\begin{matrix} \frac{1}{2} a_{1} t^{2} + z_{0}, & t \leq t_{1} \\ \frac{1}{2} a_{2} {(t - t_{t u r n})}^{2} + z_{m a x}, & t_{1} < t \leq t_{t u r n} \\ \frac{1}{2} a_{3} {(t - t_{t u r n})}^{2} + z_{m a x}, & t_{t u r n} < t \leq t_{2} \\ \frac{1}{2} a_{4} {(t - T_{s t e p})}^{2} + z_{1}, & t > t_{t u r n} \end{matrix}

(4)

where

z_{m a x} = \max {z_{0}, z_{1}} + h_{t r a j},

(5)

t_{t u r n} = {sat}_{0.25 T_{s t e p}}^{0.5 T_{s t e p}} (T - \frac{2 (z_{1} - z_{m a x})}{V_{l o w e r}}),

(6)

where the saturation function ${sat}_{a}^{b} (x)$ is defined as

{sat}_{a}^{b} (x) = {\begin{matrix} a, & x < a \\ x, & a \leq x \leq b \\ b . & x > b \end{matrix}

(7)

a₁, a₂, a₃, a₄, t₁ and t₂ can be calculated using the smooth condition, which will not be detailed here.

Figure 4.

Vertical displacement component of the swing foot trajectory

4. Compliant Adaptation of Landing Foot

Theoretically, the compliance control should be applied to all the six Dimensions of Freedom (DOF) of the landing foot in order to avoid slipping between the sole and the ground. In practice, as the horizontal displacement and the yaw rotation can be ignored during the landing process, the compliance control is only used for the roll, the pitch rotations and the vertical displacement.

4.1. Compliance control of roll and pitch rotations

There are two major considerations in choosing the compliance control method for the landing foot. First, the landing foot is supposed to hold its pose when there is no difference between the measured and the reference torque, which implies that the controller should have zero steady state error. Second, the low-pass filter for force/torque sensor signal has a larger phase delay for high frequency components than that for low frequency components. Hence a controller with low-pass property will help to reinforce the closed-loop stability by suppressing high frequency oscillations. The integral property of the damping controller makes it conform to the two requirements above. Therefore the damping control method is used to achieve compliance in the roll and the pitch rotations in this paper.

Given that the reference torque is set to zero, the angular velocity is calculated as

\dot{θ} (t) = \frac{1}{D} M^{m e a} (t),

(8)

where D is the given damping coefficient, and $M^{m e a}$ is the measurement of the torque corresponding to the rotation. Subscripts indicating roll or pitch are omitted in this section for convenience.

There are two remaining issues to be discussed: 1) how to choose an appropriate damping coefficient D and 2) the terminate condition for the compliance control.

4.1.1. Choosing damping coefficient

Ideally, the adaptation always performs better with a smaller D, which makes the foot more compliant. However, when taking time delay into consideration, the system will become divergent when D is too small. Thus there is a lower limit for D.

Assume that the contact between the sole and the terrain is linear elastic with a spring constant K. And use the notation τ for the overall time delay of the control loop. The closed-loop system can then be model as

D \dot{θ} (t) = - K θ (t - τ) .

(9)

Discretize Eq. (9) with sampling time T_s and the assumption $τ = N T_{s}$ , $N \geq 1$ . We have

D \frac{1 - z^{- 1}}{T_{s}} θ (z^{- 1}) = - K z^{- N} θ (z^{- 1}) .

(10)

The characteristic equation of Eq. (10) is

D z^{N} - D z^{N - 1} + K T_{s} = 0 .

(11)

Since the characteristic polynomial is continuous about D, and we know that the system can be stable with a big enough D or unstable with a small enough D, there exists a certain $D_{m i n}$ corresponding to the critical stable condition, which means the system is stable when and only when $D D_{m i n}$ . $D_{m i n}$ should be the maximum D for Eq. (11) to have roots on the unit cycle [23]

D_{m i n} = \max {D | \exists ω \in R, D e^{j n ω} - D e^{j (n - 1) ω} + K T_{s} = 0} .

(12)

The solution of Eq. (12) is

D_{m i n} = \frac{K τ}{2 N \sin \frac{π}{4 N - 2}} = \frac{K T_{s}}{2} \csc \frac{π}{4 N - 2}, N \in N^{+} .

(13)

It can be easily proven that

N_{0} \sin \frac{π}{4 N_{0} - 2} < \lim_{N \to \infty} N \sin \frac{π}{4 N - 2} = \frac{π}{4}, \forall N_{0} \in N^{+} .

(14)

From Eq. (13) (14), we have

D_{m i n} < \frac{2}{π} K τ, \forall N \in N^{+} .

(15)

Hence we choose D conservatively with the criterion

D > \frac{2}{π} K τ \approx 0.637 K τ .

(16)

The value of τ can be obtained from the mechanism model of the system. The value of K varies under different contact conditions. Obviously when the foot makes a full contact with an even floor, K reaches its maximum value $K_{e v e n}$ , which can be identified through shaking the foot on even floor with small amplitude. When the contact is confined to one side of the sole, it can be assumed that $K \leq 0.5 K_{e v e n}$ . Therefore the compliance control should be applied in stages with different damping coefficients.

4.1.2. Stages and terminate conditions

Ideally, the adaptation completes when the measured torque $M^{m e a}$ crosses zero. However, for two reasons the compliance control should continue in the actual situation: 1) the foot has already overreached the equilibrium position because of the time delay, and 2) the actual pose of the foot will shift for a short interval because of the dynamic deformation of the linkages. Hence the compliance control of the roll/pitch rotation should be applied in two stages with corresponding switch/terminate condition:

The damping coefficient D is set to be $D_{1} = 0.5 D_{2}$ , where D₂ is the damping coefficient value applied in the second stage. The switch condition to the next stage is that the measured torque $M^{m e a}$ crosses zero.

D is set to a certain value $D_{2} > 0.637 K_{e v e n} τ$ . The terminate condition is that the measured vertical force $F_{z}^{m e a}$ has exceeded a threshold value $F_{f i r m}$ for a time interval $T_{f i r m}$ .

4.2. Feedforward and feedback control of vertical motion

The vertical motion is related to the roll and the pitch rotations during the adaptation, as the contact point about which the foot rotates rarely falls at the center of the sole. A feedforward compensation is therefore designed to improve the performance. The overall control law is

z (t) = z_{f f} (t) + z_{f b} (t),

(17)

where $z_{f f}$ is the feedforward component and $z_{f b}$ is the feedback component.

4.2.1. Feedforward controller

To simplify the discussion, relation between the vertical motion and the rotation along one single axis is studied first, as shown in Figure 5.

Figure 5.

Relation between the vertical motion and the rotation of the landing foot

With the sole in continuous contact with the ground, and the derivative of $L_{C o P} (t)$ ignored, we have

V_{z} (t) = ω (t) L_{C o P} (t) \cos θ (t) .

(18)

Then the descending distance since the moment $t_{t o u c h}$ when the contact began should be

Δ z (t) = \int_{t_{t o u c h}}^{t} ω (t) L_{C o P} (t) \cos θ (t) d t .

(19)

However, time delay should be taken into consideration or Eq. (19) will almost certainly overestimate the descending distance. The measured COP position $L_{C o P}^{m e a}$ comes after a time delay $τ_{m e a}$ , while the commanded angle $θ^{s p}$ is followed with a actuation delay $τ_{a c t}$ . Hence Eq. (19) becomes

Δ z (t) = \int_{t_{t o u c h}}^{t} ω^{s p} (t - τ_{a c t}) L_{C o P}^{m e a} (t + τ_{m e a}) \cos θ^{s p} (t - τ_{a c t}) d t .

(20)

The right hand side of Eq. (20) contains information from the future. Approximations are needed to estimate the descending distance online. Since the overall time delay $τ = τ_{m e a} + τ_{a c t}$ , and $\cos θ \approx 1$ as θ is usually quite small, we have the estimation

Δ z^{e s t} (t) = \int_{t_{t o u c h}}^{t} ω^{s p} (t - τ) L_{C o P}^{m e a} (t) d t .

(21)

Similarly, when considering both the roll and the pitch rotations,

\begin{array}{l} V_{z}^{e s t} (t) = [\begin{matrix} 0 0 1 \end{matrix}] \cdot [- \vec{ω} (t - τ) \times {\vec{p}}_{C o P} (t)] \\ = [\begin{matrix} 0 0 - 1 \end{matrix}] \cdot ([\begin{matrix} ω_{x} (t - τ) \\ ω_{y} (t - τ) \\ 0 \end{matrix}] \times [\begin{matrix} p_{x} (t) \\ p_{y} (t) \\ p_{z} (t) \end{matrix}]) \\ = ω_{y} (t - τ) p_{x} (t) - ω_{x} (t - τ) p_{y} (t), \end{array}

(22)

where $\vec{ω}$ is the commanded rotation vector and ${\vec{p}}_{C o P}$ is the measured COP position vector. Superscripts indicating measurement or command are omitted for clarity. Therefore the feedforward component is

\begin{array}{l} z_{f f} (t) = \int_{t_{t o u c h}}^{t} V_{z}^{e s t} (t) d t \\ = \int_{t_{t o u c h}}^{t} [ω_{y} (t - τ) p_{x} (t) - ω_{x} (t - τ) p_{y} (t)] d t . \end{array}

(23)

It will be mentioned in the next section that Eq. (19) also constitutes part of the terrain estimator.

4.2.2. Feedback controller

The damping control method is employed in the feedback component of the vertical motion control, just like in the control of the roll and the pitch rotations. The difference is the reference vertical force $F_{z}^{s p} (t) > 0$ , in order to maintain the foot in contact with the ground. The control law is

{\dot{z}}_{f b} (t) = \frac{1}{D_{z}} [F_{z}^{m e a} (t) - F_{z}^{s p} (t)],

(24)

where

\begin{array}{l} F_{z}^{s p} (t) = \frac{F_{f i r m} - F_{z}^{m e a} (t_{t o u c h})}{T_{t o u c h}} (t - t_{t o u c h}) \\ + F_{z}^{m e a} (t_{t o u c h}), t \geq t_{t o u c h}, \end{array}

(25)

where $T_{t o u c h}$ is a given time interval. The choosing of D_z follows the similar rule in Eq. (16). And the identification of the vertical spring constant K_z will be discussed in the next section.

5. Correction and Adjustment

As mentioned above, the pose of the landing foot relative to the body calculated through the forward kinematics is not reliable due to the dynamic deformations in the linkages. Meanwhile the pose of the body in the world coordinates is not desired. Thus if holding the commanded pose of the landing foot at the time $t_{f i r m}$ when the adaptation has finished, the robot will deviate from its desired movement and fall down in the following steps. Therefore a more accurate estimation of the terrain is needed to provide the corrected target pose for the supporting foot. The corresponding correction process also needs to be specified.

The height of the COM should adjust to the variation of the terrain in order to maintain the vertical distance between the COM and the terrain. This is the only adjustment of the COM trajectory in this research.

5.1. Terrain estimation

The deviation between the calculated and the actual foot pose relative to the body obeys different rules in different phases:

During the supporting phase and the flight phase, including the moment $t_{t o u c h}$ when the flight phase ends, the deviation concentrates on the vertical displacement, and is proportional to the vertical pressure on the leg.

During the touching phase, the deviation can still be deemed to exist only in the vertical displacement. However it is no longer proportional to the pressure because the deformations of the linkages is dynamic for a period after the impact.

Therefore the height estimation is the key point in the terrain estimation. The basic idea is to use the pose at $t_{t o u c h}$ calculated through the kinematics as an initial value, and the descending distance calculated from the contact force/torque information as the increment value, with the body attitude taken into consideration.

Here, we assume the left foot to be the supporting foot and the right foot to be the landing foot. The deviation of the body from its desired pose at $t_{t o u c h}$ , as shown in Figure 6, is equivalent to rotating the robot about the left foot by the body attitude angle measured by the IMU [10], then lowering the body by the length that the left leg is compressed.

Figure 6.

Deviation of the body from its desired pose

The actual position of the right foot in the world coordinates is

\begin{array}{l} ​^{W} {\vec{p}}_{R_{a c t}} =^{W} {\vec{p}}_{L_{a c t}} +^{W} R_{B_{a c t}} ​^{B_{a c t}} {\vec{p}}_{a c t} \\ =^{W} {\vec{p}}_{L_{a c t}} +^{W} R_{B_{a c t}} (^{B_{a c t}} {\vec{p}}_{R_{a c t}} -^{B_{a c t}} {\vec{p}}_{L_{a c t}}), \end{array}

(26)

where $^{W} {\vec{p}}_{L_{a c t}}$ was obtained in the previous step, and $^{W} R_{B_{a c t}}$ is calculated from the IMU sensor data. The actual foot positions relative to the body remain to be acquired. With the analysis of the deviation above, we have

{\begin{matrix} ​^{W} R_{B_{a c t}} (^{B_{a c t}} {\vec{p}}_{L_{a c t}} -^{B_{s p}} {\vec{p}}_{L_{s p}}) = [\begin{matrix} 0 \\ 0 \\ K_{z} F_{z}^{L} \end{matrix}], & ​ \\ ​^{W} R_{B_{a c t}} (^{B_{a c t}} {\vec{p}}_{R_{a c t}} -^{B_{s p}} {\vec{p}}_{R_{s p}}) = [\begin{matrix} 0 \\ 0 \\ K_{z} F_{z}^{R} \end{matrix}], & ​ \end{matrix}

(27)

where $^{B_{s p}} {\vec{p}}_{L_{s p}}$ and $^{B_{s p}} {\vec{p}}_{R_{s p}}$ are calculated through the kinematics using joint angles. Then the actual position of the landing foot can be further written as

​^{W} {\vec{p}}_{R_{a c t}} =^{W} {\vec{p}}_{L_{s p}} + (^{W} R_{B_{a c t}} ​^{B_{s p}} {\vec{p}}_{R_{s p}} + [\begin{matrix} 0 \\ 0 \\ K_{z} F_{z}^{R} \end{matrix}])

- (​^{W} R_{B_{a c t}} ​^{B_{s p}} {\vec{p}}_{L_{s p}} + [\begin{matrix} 0 \\ 0 \\ K_{z} F_{z}^{L} \end{matrix}])

=^{W} {\vec{p}}_{L_{s p}} +^{W} R_{B_{a c t}} (^{B_{s p}} {\vec{p}}_{R_{s p}} -^{B_{s p}} {\vec{p}}_{L_{s p}})

+ [\begin{matrix} 0 \\ 0 \\ K_{z} (F_{z}^{R} - F_{z}^{L}) \end{matrix}] .

(28)

The actual position of the landing foot in the world coordinates at the moment $t_{t o u c h}$ is then obtainable with Eq. (28).

Method similar to Eq. (19) is used to obtain the descending distance of the landing foot, with additional consideration of the body rotations

\begin{array}{l} ​^{W} V_{z}^{e s t} (t) = [\begin{matrix} 0 0 1 \end{matrix}] \cdot [-^{W} {\vec{ω}}_{R_{a c t}} (t) \times^{W} {\vec{p}}_{C o P} (t)] \\ = [\begin{matrix} 0 0 - 1 \end{matrix}] \cdot {[^{W} {\vec{ω}}_{B_{a c t}} (t) +^{W} R_{B_{a c t}} {(t)}^{B_{s p}} {\vec{ω}}_{R_{s p}} (t)] \\ \times [​^{W} R_{B_{a c t}} {(t)}^{B_{s p}} R_{R_{s p}} {(t)}^{R_{a c t}} {\vec{p}}_{C o P} (t)]} . \end{array}

(29)

With approximations and the time delay, Eq. (29) can be further written as

\begin{array}{l} ​^{W} V_{z}^{e s t} (t) \approx [\begin{matrix} 0 0 - 1 \end{matrix}] \cdot {[^{W} {\vec{ω}}_{B_{a c t}} (t - τ) +^{B_{s p}} {\vec{ω}}_{R_{s p}} (t - τ)] \\ \times^{R_{a c t}} {\vec{p}}_{C o P}^{m e a} (t)} . \end{array}

(30)

Thus the position of the landing foot at the moment $t_{f i r m}$ is

​^{W} {\vec{p}}_{R_{a c t}} (t_{f i r m} {) =}^{W} {\vec{p}}_{R_{a c t}} (t_{t o u c h}) + [\begin{matrix} 0 \\ 0 \\ {\int_{t_{t o u c h}}^{t_{f i r m}}}^{W} V_{z}^{e s t} (t) d t \end{matrix}],

(31)

where $^{W} {\vec{p}}_{R_{a c t}} (t_{t o u c h})$ and $^{W} V_{z}^{e s t} (t)$ are calculated using Eq. (28) and Eq. (30).

Besides the position estimation, the attitude of the terrain also needs to be estimated. The rotation matrix of the landing foot in the world coordinates at moment $t_{f i r m}$ can be calculated with

​^{W} R_{R_{a c t}} (t_{f i r m} {) =}^{W} R_{B_{a c t}} {(t_{f i r m})}^{B_{s p}} R_{R_{s p}} (t_{f i r m}) .

(32)

The terrain estimation is then achieved using Eq. (31) (32).

The vertical spring constant K_z is obtained through identification. Notice that the vertical elasticity comes from both the contact with the ground and the compression of the leg. Hence K_z is smaller than the elastic coefficient calculated from the Young's modulus of the contact materials. Let the robot walk on an even floor with the original open-loop gait. The actual height of the foot at $t_{t o u c h}$ must be

​^{W} {\vec{p}}_{R_{a c t}} = [\begin{matrix} ⋮ \\ 0 \end{matrix}] .

(33)

However, the height calculated through kinematics would be different because of the vertical elasticity. With Eq. (27) and Eq. (33), we have

K_{z} = \frac{[\begin{matrix} 001 \end{matrix}] (^{W} {\vec{p}}_{L_{s p}} +^{W} R_{B_{a c t}} ​^{B_{s p}} {\vec{p}}_{s p})}{F_{z}^{L} - F_{z}^{R}} .

(34)

The mean value of the results from multiple trials is then used as K_z in both the choosing of the vertical damping efficient D_z and the terrain height estimation.

5.2. Correction

With the actual position $^{W} {\vec{p}}_{R_{a c t}}$ and the actual attitude $^{W} R_{R_{a c t}}$ obtained, the commanded position $^{W} {\vec{p}}_{R_{s p}}$ and the commanded attitude $^{W} R_{R_{s p}}$ should be corrected to the actual values, so that the body can be pushed back to its desired trajectory. Define the actual values of the landing foot pose as the reference values

(\begin{matrix} {\vec{p}}_{r e f} =^{W} {\vec{p}}_{R_{a c t}}, & ​ \\ R_{r e f} =^{W} R_{R_{a c t}} . & ​ \end{matrix}

(35)

To avoid discontinuities in velocities, a same second-order correction process $G (s)$ is applied in each component of the landing foot pose

G (s) = \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}} .

(36)

The damping coefficient ζ is set to 1 as overshoot is undesired. The unit step response of a second-order system under critical damping exceeds 0.9 after the moment $6 / ω_{n}$ . In order to complete the correction in a time interval $T_{s e t}$ , we choose

ω_{n} = \frac{6}{T_{s e t}} .

(37)

Substitute Eq. (37) into Eq. (36), we have

G (s) = \frac{36}{T_{s e t}^{2} s^{2} + 12 T_{s e t} s + 36} .

(38)

5.3. Height adjustment

The desired COM height $^{W} z_{B_{s p}}$ should change according to the variation of the terrain height. It has been proven that when the COM of an inverted pendulum which is supported at the original point, moves within a constraint plane

z = k_{x} x + k_{y} y + z_{c},

(39)

the horizontal motion of this inverted pendulum is the same with that of a linear inverted pendulum. Hence a height adjustment strategy similar to the classic stair climbing algorithm is designed, as shown in Figure 7, with the difference that the terrain height is unknown before the landing of the foot. Once the landing foot has made a firm contact with the terrain and becomes the supporting foot, the robot begins to move its COM within the newly calculated constraint plane, until the COM passes the new supporting foot in the forward direction, after which the COM will move within the horizontal plane as the trend of the terrain in the next step is still unknown.

Figure 7.

COM height adjustment according to the terrain

6. Experimental Results

Figure 8 shows the humanoid robot Kong and the uneven terrain used in our experiments. The full-size humanoid robot Kong is 161 cm in height and 53 kg in weight with a total DOF of 14 [24]. A six-dimensional force/torque sensor is mounted in each foot. And an IMU is mounted under the waist between two hips, where the COM is considered to be. The amplifiers communicate with the upper computer through the CAN network. The uneven terrain was built by attaching a 16 mm thick hard plank to a common floor.

Figure 8.

The humanoid robot Kong and the uneven terrain used in the experiments

We validated our method by walking experiments with Kong. The walking speed was 0.225 kmph, while the step length was 5 cm. The total step count in each experiment was 12. The first two steps in each experiment didn't use the method proposed in this paper, as to ensure a reliable startup.

Walking experiment on the uneven terrain with the proposed methods was implemented to verify the effectiveness of the methods, as shown in Figure 9. In addition, the walking experiments on an even floor with or without the proposed methods have been compared. The results of walking on the uneven floor without the proposed methods are not shown because the robot fell almost immediately after stepping on the obstacle. The body attitude angles in these experiments are shown in Figure 10 as an indicator of walking stability. The body attitude angles in all these experiments fell within about $\pm 2^{\circ}$ . When walking on the uneven terrain, the magnitude of the angles increased only slightly comparing to that when walking on even floor. And when walking on the even floor, the results using or not using the proposed methods are similar, with the magnitude of the pitch angle being smaller when using the proposed methods. Thus, it can be claimed that our new methods make the robot capable of walking on uneven terrains stably, without degrading the performance on even floors.

Figure 9.

Snapshots of the walking experiment on the uneven terrain. Multiple local terrain cases were included.

Figure 10.

Body attitude angles of walking on even or uneven terrain with or without the methods proposed in this paper. The start time of the uneven terrain intervention is the moment when the swing foot first touched the uneven terrain. And the end time of the uneven terrain intervention is the moment when the supporting foot left the uneven terrain for the last time.

To examine the effectiveness of the foot compliance, the vertical force F_z on each foot in the experiments with or without the proposed methods are compared, as shown in Figure 11. Only results of walking on the even floor are included in this comparison for the consistency of contact conditions. The contacts started earlier when using the proposed methods because the new swing foot trajectories aimed lower than the traditional trajectories. The vertical force control decreased the first peak in every step significantly, and steadied the vertical force during the adaptations, relieving the “pre-landing” phenomenon. As a result of this impact absorption, F_z during the supporting phase became steadier than that in the result without the new methods. Hence, the foot compliance enhanced the walking stability by reducing the impact, and guaranteed the terrain estimation to be effective by maintaining the landing foot in continuous contact with the ground.

Figure 11.

Vertical force on each foot when walking on even floor with or without the proposed methods. Only part of the time axis is shown for the clarity of the figure.

We shall then verify the accuracy of the terrain estimation. As shown in Figure 12, the terrain attitude angle estimation error usually fell within $\pm 1^{\circ}$ with very few exceptions, while the terrain height estimation error usually fell within $\pm 2$ mm. The terrain estimation is accurate enough to guarantee the walking stability while maintaining the original horizontal COM trajectory.

Figure 12.

Estimated and referenced values of the local terrain at each foot landing point. Only part of the time axis is shown for the clarity of the figure. The lines represent the poses of the feet. The markers represent the referenced values of the terrain, whose abscissas correspond the moments when the adjustment finished in each step. Therefore the estimation errors are represented by the vertical distance between the lines and the corresponding markers.

Reference and measured ZMP of walking on even or uneven terrain with the proposed methods are shown in Figure 13. The ZMP following errors are noticeable, as only the offline COM trajectory generated by the basic preview control method was applied, while the robot faced the problems like the modelling error and the oscillation caused by the elasticity of the linkages. Stable walking under such condition shows that the proposed methods are not picky about the high-level controller or the hardware platform. It can also be noted that the measured ZMP of walking on the even and on the uneven terrain differed little except for the time shifting caused by the different landing conditions. Therefore the ZMP trajectory can be expected to be maintained under different terrain conditions.

Figure 13.

Reference and measured ZMP of walking on even or uneven terrain with the methods proposed in this paper. The start and end time of the uneven terrain intervention are defined in the same way as in Figure 10.

7. Conclusions and Future Work

In this paper, we propose a control algorithm that enables humanoid robots to walk on uneven terrains. The major difference between our algorithm and previous works is that our algorithm is based on an online terrain estimation, which brings the benefit that only minimum modification to the walking gait is required. The instant terrain estimation is achieved by using information from different sensors at different stages, considering that the forward kinematics is not always reliable because of the deformations in the linkages. In addition, a new swing foot trajectory and a compliant terrain adaptation method for the landing foot are designed to cope with various terrain conditions. We validated our algorithm by walking experiments on the full-size humanoid robot Kong. The results show that the performance of the robot when walking on the uneven terrain is as good as that when walking on the even floor. The effectiveness of the compliant terrain adaptation and the accuracy of the terrain estimation were also verified.

The allowed terrain height variation range $[H_{0} - Δ H_{m i n}, H_{0} + Δ H_{m a x}]$ is currently limited by the hardware performance, which stops the swing foot from accomplishing more adaptable trajectories. We believe our method can handle more challenging terrains on our new hardware platform in the future. And there are still some to improve within the framework of our methods. Slipping between the sole and the terrain is inevitable while walking, which can change the contact conditions especially when causing the toe or the heel to fall from the edge of an obstacle. Methods that recognize such terrain changes and re-adapt the foot to the terrain are therefore required for a better performance. Furthermore, strategies to re-balance the robot should be added to the framework to improve the robustness of the control system.

Footnotes

8. Acknowledgements

This research is supported by the National Foundation of China (Grant No. 51405430, Grant No. 61473258 and Grant U1509210).

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