Modelling and control for a bipedal robot on slopes

Abstract

This paper presents a systematic trajectory generation method for a bipedal robot walking on slopes. A suitable offline walking pattern based on an inverted pendulum model is designed and the planning is parameterized by step lengths, step period and other walking parameters. Meanwhile, a slight unevenness of a slope can cause serious instability for bipedal walking robots. Therefore, this paper also proposes an online control algorithm for a bipedal robot even walking on the unevenness slope, and the robot can adapt to the floor conditions. The control algorithm includes a landing time controller, landing direction controller, zero moment point (ZMP) regulation controller and attitude correction controller. During the process, accurate attitude information for these controllers is achieved through an adaptive filtering method and the ZMP position is measured through force sensing register sensors, which are attached to the robot’s feet. Finally, the experiment is carried out on a SCUT-I humanoid robot. The result proves that the method described in this paper can successfully control a robot walking on slopes.

Keywords

Bipedal robot SCUT-I slope walking control

Introduction

Bipedal robots are more flexible compared with wheeled robots. Many bipedal humanoid robots have been developed, such as WABIAN (Yamaguchi et al., 1998), Asimo (Hirai et al., 1998), HRP (Akachi et al., 2005), KHR (Kim et al., 2007) etc.

Some floors that appear to be flat have local and global inclinations to some degree. Bipedal robots are not very stable during locomotion, so research on how bipedal robots walk on slopes is needed. Some researchers have done some work on this.

Zhou et al. (2004) formulated gait synthesis of humanoid bipedal locomotion as an optimization problem with consideration of some constraints, e.g. zero moment point (ZMP) constraints for dynamically stable locomotion, internal forces constraints for smooth transition and geometric constraints for walking on slopes. Huang et al. (2008) proposed a motion pattern generator for slope walking in 3D dynamics using preview control of a ZMP. Yet these researchers mainly considered offline gait planning of walking on slopes, and feedback controllers are not designed to adapt to the changes in the floor.

Kim et al. (2007) designed a walking control algorithm for a bipedal humanoid robot stably walking on an inclined floor, and the performance of the algorithm was demonstrated in walking experiments using the KHR-2. Suwanratchatamanee et al. (2010) presented a tactile sensing system for humanoid robots walking on slopes autonomously, but how the procedure of bipedal humanoid robots walking on slopes is parameterized and how to fuse and calculate the robot sensor’s data were not mentioned in these papers.

A systematic trajectory generation method is proposed in this paper for a bipedal robot walking on slopes. Firstly, a suitable offline walking pattern based on an inverted pendulum model (IPM) is designed and the planning of a bipedal robot walking on slopes is parameterized. Secondly, an online control algorithm for a bipedal robot walking on slopes is proposed. The robot’s attitude sensors information is fused to improve the accuracy of the robot’s attitude. ZMP position is calculated based on force sensing register (FSR) sensors attached to the sole of the robot’s feet. Finally, the stage of a bipedal robot walking on slopes is divided according to FSR sensors, and some feedback controllers are designed to control bipedal humanoid robots stably walking on slopes. The method of a bipedal robot walking on slopes in this paper is shown in Figure 1.

Figure 1.

The method of a bipedal robot walking on slopes.

This paper is organized as follows. In the next section, the SCUT-I bipedal humanoid robot, the robot experimental platform, is introduced. Then, an offline gait plan is designed based on IPM (Kajita et al., 2001) in the x- and y-axes, and the gait plan of the robot walking on slopes is parameterized based on some walking parameters. An adaptive filtering method is presented to obtain accurate attitude information and the ZMP position can be calculated based on eight FSR sensors attached to the robot’s feet. Then, the landing time controller, landing direction controller, ZMP regulation controller and attitude correction controller are designed. In this section, the walking process is divided according to the values from the eight FSR sensors and the controller action time that is planned. When the controller is not active, its controller’s parameters are all zero. The performance of the walking control algorithm is assessed and demonstrated through the experiment, and finally, conclusions and the future work are proposed.

Bipedal humanoid robot platform, SCUT-I

The SCUT-I is a bipedal humanoid robot, shown in Figure 2(a). It is used as an experimental platform in this paper. The robot has a height of 582 mm, and only weights 3.9 kg, including batteries. The robot has 21 servo motors: six in each leg, three in each arm, two in the head and one in the trunk. The three-degree-of-freedom (3DoF) orthogonal hip joint and 2DoF orthogonal ankle joint altogether make it possible for the bipedal robot to walk, and the joint in the trunk allows the robot to execute various kinds of movements, such as kicking, goal keeping, getting up from the ground etc.

Figure 2.

SCUT-I robot and model: (a) SCUT-I robot; (b) SCUT-I robot model.

SCUT-I bipedal humanoid robot model

The robot model is shown in Figure 2(b), in which the arms trajectories are not taken account of, and where m_i represents the ith part mass and l_i represents the ith part length. The parameters of the robot model are listed in Table 1. The mathematical model of the real SCUT-I bipedal humanoid robot is built. A forward kinematics (FK) mathematic model is based on the screw theory (Kajita, 2005), and an inverse kinematics (IK) mathematic model is based on the geometry relationship in SCUT-I. The robot’s joint angles can be calculated according to the trajectory of hip and ankle.

Table 1.

Structure parameter of SCUT-I biped humanoid robot.

Parameter	Value	Parameter	Value	Parameter	Value	Parameter	Value
m ₁	1.55 kg	m ₈	0.01 kg	l ₁	60 mm	l ₈	0 mm
m ₂	0.01 kg	m ₉	0.30 kg	l ₂	0 mm	l ₉	0 mm
m ₃	0.30 kg	m ₁₀	0.05 kg	l ₃	0 mm	l ₁₀	135 mm
m ₄	0.05 kg	m ₁₁	0.20 kg	l ₄	135 mm	l ₁₁	135 mm
m ₅	0.20 kg	m ₁₂	0.30 kg	l ₅	135 mm	l ₁₂	0 mm
m ₆	0.30 kg	m ₁₃	0.30 kg	l ₆	0 mm	l ₁₃	12 mm
m ₇	0.30 kg			l ₇	12 mm

Meanwhile, a ZMP (Vukobratovic, 2004) stability equation of the SCUT-I robot involving momentum, angular momentum, ZMP, velocity and acceleration is established and used as the mathematical basis of bipedal walking stability analysis.

SCUT-I bipedal humanoid robot sensor system

The bipedal robot’s attitude in rolling and pitching directions is needed, which is important to control robot to walk stably. Therefore, attitude sensors need to be mounted on the robot, which can measure the rolling and pitching directions. Acceleration sensor ADXL204 (Analog Device Company, 2006) and gyroscope IDG-300 (InvenSence Device Company, 2007) are mounted at the torso of the SCUT-I robot, as shown in Figure 3(a).

Figure 3.

SCUT-I robot sensor system: (a) attitude sensors; (b) force sensing register (FSR) sensor.

It is also important to learn how to obtain the force information of the robot feet during the walking process. In this paper, four FSR sensors are attached to every foot (Figure 3b). So the condition of feet contacting with the ground can be obtained and the ZMP position according to the value of FSR sensors can be calculated.

Offline gait generation of bipedal robot

An offline gait trajectory is firstly designed to make the robot walk. The gait planning of the bipedal robot walking on slopes is parameterized based on step length s, walking cycle $T_{sup}$ , height of the centre mass $h_{c}$ , height of foot $h_{foot}$ , double-support time $T_{db}$ , rising foot time $T_{up}$ and landing foot on the ground time $T_{down}$ , and the trajectory of hip and ankle can be achieved according to the parameters.

The planning of bipedal robots walking on slopes

The slope-walking gait can be considered a sequence of steps. The walking cycle is composed of two phases: 1) a double-support phase (short for DSP) and 2) a single-support phase (short for SSP).

In DSP, both feet are fixed on slopes. The robot accelerates from standing still, and the centre of gravity (COG) moves in the forward direction. At this time, hip trajectory is designed through a sixth-order polynomial and the ankle trajectory is unchanged.

In SSP, only one foot is stationary on the ground, and the other foot swings from the rear position to the front position. This includes three phases: 1) a rising foot phase, 2) a swinging foot phase and 3) a landing foot phase. At this time, hip trajectory is generated according to the COG trajectory, which is based on IPM in the x- and y-axes, and the ankle trajectory is planned base on the position of the landing point on the ground and sine curve.

The plan of the robot walking on slopes is shown in Figure 4. In the figure, the left leg is the supporting leg and the right leg is the swinging leg from position $P^{n - 2}$ to position $P^{n}$ .

Figure 4.

The planning of the robot walking on slopes.

The key points of the cyclical walking period are defined as $(n - 1) T_{sup}$ , $(n - 1) T_{sup} + T_{db}$ , $(n - 1) T_{sup} + T_{db} + T_{up}$ and $(n - 1) T_{sup} + T_{db} + T_{up} + T_{over}$ , where $T_{over}$ is the swinging foot time and $T_{down}$ = $T_{sup} - T_{db} - T_{up} - T_{over}$ .

According to the plan of the robot walking on slopes, s, $T_{sup}$ , $T_{db}$ , $T_{up}$ , $T_{down}$ , $h_{c}$ and $h_{foot}$ are selected to represent the walking slope movement. It is shown by Equation (1).

P_{robot} = f (s, T_{sup}, T_{db}, T_{up}, T_{down}, h_{c}, h_{foot})

(1)

The procedure of the gait planning of the robot walking on slopes is shown in Figure 5.

Figure 5.

The procedure of the bipedal humanoid robot walking on slopes.

Trajectory generation of the robot walking on slopes

If ankle and hip trajectories of swinging leg are determined, all joint trajectories can be calculated by IK. Therefore, the gait of the robot walking on slopes can be planned through the swinging ankle joint and hip joint trajectory.

A schematic diagram of the robot walking on a slope is shown in Figure 6, where $p_{2}$ , $p_{8}$ , $p_{7}$ and $p_{13}$ are the co-ordinates of the hip and ankle. $θ$ is the inclination angle of the slope, and the right leg is swinging leg and the left leg is the support leg. So the right leg’s trajectory is mainly analysed and designed. When the left leg is the swinging leg, the procedure is similar.

Figure 6.

Bipedal humanoid robot walking on a slope.

Hip trajectory of swinging leg

The hip trajectory is generated according to the COG trajectory.

p_{2} = p_{c} + {[\begin{matrix} 0 & l_{1} & - Δ h_{h c} \end{matrix}]}^{T}

(2)

where $p_{c}$ is the COG co-ordinate and $Δ h_{hc}$ is the distance between the COG and hip in z-axis.

p_{c} = {[\begin{matrix} x_{c} & y_{c} & z_{c} \end{matrix}]}^{T} = {\begin{matrix} [\begin{matrix} x_{c_dbl} & y_{c_dbl} & z_{c} \end{matrix}] & (n - 1) T_{\sup} \leq t \leq (n - 1) T_{\sup} + T_{d b} \\ [\begin{matrix} x_{c_sgl} & y_{c_sgl} & z_{c} \end{matrix}] & (n - 1) T_{\sup} + T_{d b} \leq t \leq n T_{\sup} \end{matrix}

${[\begin{matrix} x_{c_dbl} & y_{c_dbl} & z_{c} \end{matrix}]}^{T}$ is the COG co-ordinate in DSP. The $x_{c_dbl}$ and $y_{c_dbl}$ trajectories are designed through a sixth-order polynomial; $[\begin{matrix} x_{c_sgl} & y_{c_sgl} & z_{c} \end{matrix}]$ is the COG co-ordinate in SSP. The $x_{c_sgl}$ and $y_{c_sgl}$ trajectories are designed through a 3D IPM. The $z_{c}$ trajectory is designed for walking on slopes and shown as follows:

z_{c} = x_{c} * \tan θ + h_{c}

Ankle trajectory of swinging leg

The ankle trajectory is planned based on the position of the landing point on the ground. The trajectory in the z-axis is shown by a sine curve. The landing point can be planned according to the walking target position, and adjusted according to the current walking condition (Kajita, 2005). The trajectories in x-, y- and z-axes are shown as follows:

x_{p_{7}} (t) = {\begin{matrix} x_{p_{7}}^{P^{n - 2}} (n - 1) T_{sup} \leq t \leq (n - 1) T_{sup} + T_{db} \\ \frac{x_{p_{7}}^{P^{n}} - x_{p_{7}}^{P^{n - 2}}}{T_{sup} - T_{db}} * (t - (n - 1) T_{sup} - T_{db}) + x_{p_{7}}^{P^{n - 2}} (n - 1) T_{sup} + T_{db} \leq t \leq n T_{sup} (3) \end{matrix}

(3)

y_{p_{7}} (t) = {\begin{matrix} y_{p_{7}}^{P^{n - 2}} (n - 1) T_{sup} \leq t \leq (n - 1) T_{sup} + T_{db} \\ \frac{y_{p_{7}}^{P^{n}} - y_{p_{7}}^{P^{n - 2}}}{T_{sup} - T_{db}} * (t - (n - 1) T_{sup} - T_{db}) + y_{p_{7}}^{P^{n - 2}} (n - 1) T_{sup} + T_{db} \leq t \leq T_{sup} \end{matrix} (4)

(4)

z_{p_{7}} (t) = {\begin{matrix} z_{p_{7}}^{P^{n - 2}} (n - 1) T_{sup} \leq t \leq (n - 1) T_{sup} + T_{db} \\ h_{foot} * \sin (\frac{t - (n - 1) T_{sup} - T_{db}}{2 T_{up}} π) + l_{7} + x_{p_{7}} (t) * \tan θ (n - 1) T_{sup} + T_{db} \leq t \leq (n - 1) T_{sup} + T_{db} + T_{up} \\ h_{foot} + l_{7} + x_{p_{7}} (t) * \tan θ (n - 1) T_{sup} + T_{db} + T_{up} \leq t \leq (n - 1) T_{sup} + T_{db} + T_{up} + T_{over} \\ h_{foot} * \cos (\frac{t - (n - 1) T_{sup} - T_{db} - T_{up} - T_{over}}{2 T_{down}} π) + l_{7} + x_{p_{7}} (t) * \tan θ (n - 1) T_{sup} + T_{db} + T_{up} + T_{over} \leq t \leq n T_{sup} \end{matrix}

(5)

where $x_{p}$ is defined to represent the p co-ordinate in the x-axis, and $x_{p}^{P n}$ is defined to represent the p co-ordinate in the x-axis at position $P^{n}$ . Compared with x, the definition of y and z are similar.

Sensors fusion and measurement for the robot walking on slopes

In order to improve the measurement accuracy of robot attitude, a Sage–Husa adaptive Kalman filtering method (Sage and Husa, 1969) is simplified and improved for the system, and an adaptive R is realized. Therefore, the accurate attitude information is achieved. In order to achieve condition of the feet contacting with the ground, four FSR sensors are attached to every foot and the ZMP position can be calculated.

An adaptive filtering method for improving attitude measurement accuracy

The state and observation equation of bipedal robot attitude

The state equation of bipedal robot attitude is shown as follows.

X_{k, k - 1} = Φ k, k - 1 X_{k - 1} + Γ_{k, k - 1} W_{k - 1} = [\begin{matrix} 1 & Δ t & Δ t^{2} / 2 \\ 0 & 1 & Δ t \\ 0 & 0 & 1 \end{matrix}] X_{k - 1} + [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] W_{k - 1}

(6)

The observation equation is shown as follows.

Z_{k} = H X_{k} + V_{k} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}] X_{k} + V_{k}

(7)

where $X_{k, k - 1}$ is the state variable at the time step k related to the state at the previous time step k−1. $X_{k, k - 1}$ is a vector, $X_{k, k - 1} = [{\begin{matrix} θ_{k, k - 1} & {\overset{\cdot}{θ}}_{k, k - 1} & {\overset{\cdot\cdot}{θ}}_{k, k - 1}] \end{matrix}}^{T}$ , where $θ_{k, k - 1}$ , ${\overset{\cdot}{θ}}_{k, k - 1}$ and ${\overset{\cdot\cdot}{θ}}_{k, k - 1}$ are respectively the angle, angular velocity and angular acceleration at time step k. $θ_{k, k - 1}$ is achieved by acceleration sensor and ${\overset{\cdot}{θ}}_{k, k - 1}$ is achieved by gyroscope. $Z_{k}$ is the observation vector. $Φ_{k, k - 1}$ is the state transition matrix. $Γ_{k, k - 1}$ is the process noise transition matrix. $H$ is the measurement matrix. $W_{k}$ is the system noise process modelled as a zero-mean white Gaussian random process with known covariance matrix $Q_{k}$ , $W_{k} ~ N (0, Q_{k})$ . $V_{k}$ is the observation noise process additive zero-mean Gaussian noise vector with variance $R_{k}, V_{k} ~ N (0, R_{k})$ . $Δ t$ is the interval between time step k−1 and step k.

Modified Sage–Husa adaptive Kalman filtering model for bipedal robot attitude

The Sage–Husa adaptive filtering algorithm is an improvement on the traditional Kalman filter. The value of $R$ and $Q$ would be updated in real time, which makes the filter complicated.

According to the character of bipedal robot attitude, the mean value of the observation error is r=0 and the mean value of the process error is q=0. So the traditional Sage–Husa adaptive filtering algorithm is simplified. Yet the traditional Sage–Husa algorithm cannot give estimated values when all of the dynamic noises and observation noises are unknown. Moreover, it is apt to be divergent in a high-order system. So it needs to be modified according to the character of the robot attitude.

The observation error covariance matrix $R_{k}$ is more sensitive to the filter performance for the bipedal robot attitude. So the process error covariance matrix $Q_{k}$ is set as unchangeable and $R_{k}$ can be adjusted according to the change of environment.

In the practical application, the divergence criterion of filter is designed. It is shown as follows:

{\tilde{Z}}_{k} {\tilde{Z}}_{k}^{T} > = η * tr (E [{\tilde{Z}}_{k} {\tilde{Z}}_{k}^{T}])

When the real error is larger than the ideal error $η$ times, where $η$ is repertory factor and $η \geq 1$ , $R_{k}$ needs to be adaptively modified to decrease the error.

Calculation of ZMP by FSR sensors

FSR sensors are attached to the robot’s feet, shown in Figure 3(b). The square areas under the feet are labelled during the single-supporting phase and the double-supporting phase respectively, shown in Figure 7. A, B, C and D are the positions of the FSR sensors. FW is the width of robot foot and FL is the length of robot foot. $s_{x}$ and $s_{y}$ are step length in the x- and y-axes respectively.

Figure 7.

The position force sensing register (FSR) sensor under the feet: (a) single-support phase; (b) double-support phase.

The ZMP position is calculated by Equation (8) according to the value of the FSR sensors. $p_{i}$ is the co-ordinate of the ith FSR and $f_{i}$ is the force on the ith FSR.

ZMP = \frac{\sum_{i = 1}^{N} f_{i} p_{i}}{\sum_{i = 1}^{N} f_{i}}

(8)

$Δ ZMP$ is the difference between the ZMP and support area centre, which can be calculated by Equation (9), and it is often used to show the walking stability of the bipedal robot.

Δ ZMP = ZMP - P_{M} = {[\begin{matrix} x_{ZMP} - x_{P_{M}} & y_{ZMP} - y_{P_{M}} \end{matrix}]}^{T}

(9)

Online control algorithm

Even a well-designed walking pattern cannot prevent the robot from falling as a result of the large upper body motions, vibrations of the body parts and an uneven floor. Therefore, an online walking control algorithm composed of various online controllers is essential to maintain the dynamic balance in real time. These online controllers can finely compensate for the joint trajectories.

The sensory feedback controller presented in this section consists of a landing time controller, a landing direction controller, a ZMP regulation controller and an attitude correction controller.

The proposed walking control method is based on a controller switching strategy, and thus it is important to divide the walking cycle into several walking phases.

Walking phases and controller action time

According to the offline gait trajectory, the walking cycle is divided into a DSP and an SSP, where the support foot is regarded as in full contact with the ground. Yet during the walking process of a real robot, some area of the support foot often leaves the ground. So the walking phase should be divided more minutely for a real robot. The walking process can be divide into: double-support fully contact phase, double-support not full contact phase, single-support fully contact phase and single-support not full contact phase.

In each walking cycle, suitable online controllers are activated. Figure 8 shows the walking phases and controller action time assignment.

Figure 8.

Walking phases and controller action time assignment.

Landing time controller

During the walking process of robots, the landing time of each foot is prescribed by a walking pattern design. Yet when the bipedal robot walks on a slope that is a little uneven, the swinging foot may land earlier or later than the desired landing time, which may cause a large impact force. So it is necessary to design a landing time controller to solve the problem. The landing time controller is designed in Figure 9. The controller is effective during the SSP when the landing swinging foot on the ground.

Figure 9.

Landing time controller schematic diagram.

$F_{A}'$ , $F_{B}'$ , $F_{C}'$ and $F_{D}'$ are the force of the swinging foot; $F_{value}$ is the threshold value to decide whether the foot is on the ground.

When $Δ F \geq 0$ , the robot stops the landing foot. Yet when $Δ F < 0$ , there are two conditions: 1) if landing foot phase of offline gait planning ends, a PD controller is designed, as shown in Equation (10); 2) if landing foot phase of offline gait planning does not end, the foot’s landing trajectory is based on offline gait trajectory.

Δ q_{lhp} = K_{Plhp} Δ F + K_{Dlhp} \frac{d Δ F}{dt}

(10)

where $Δ q_{lhp}$ is the adjusted quantity of the swinging hip joint, $K_{Plhp}$ and $K_{Dlhp}$ are the proportional and differential gains, and $Δ F = (F_{A}' + F_{B}' + F_{C}' + F_{D}') - F_{valve}$ .

According to the SCUT-I robot’s weight, $F_{valve}$ =3.9×9.8×0.5=19.11 N.

Landing direction controller

The robot often needs to adjust the direction of its foot to contact fully with the surface when walking on a slope. It is necessary to design a landing direction controller to solve the problem. The landing direction controller is designed in Figure 10. The controller is effective during the double-support not full contact phase.

Figure 10.

Landing direction controller schematic diagram.

The direction of the foot can be estimated by calculating the difference between the FSR sensors on the swinging foot. $Δ q_{lar}$ and $Δ q_{lap}$ are the adjusted quantities of swinging ankle joint in the rolling and pitching directions respectively. The modification is calculated as follows:

Δ q_{lar} = K_{Llar} ((F_{A}' + F_{C}') - (F_{B}' + F_{D}')) + K_{Dlar} \frac{d ((F_{A}' + F_{C}') - (F_{B}' + F_{D}'))}{dt}

(11)

Δ q_{lap} = K_{Llap} ((F_{A}' + F_{B}') - (F_{C}' + F_{D}')) + K_{Dlap} \frac{d ((F_{A}' + F_{B}') - (F_{C}' + F_{D}'))}{dt}

(12)

$K_{Llar}$ , $K_{Dlar}$ , $K_{Llap}$ and $K_{Dlap}$ are the proportional and differential gains.

ZMP regulation controller

If the actual ZMP position is controlled within a desired stable region, the robot can maintain stability well. Within a certain range of ZMP, the robot can walk stably. So a non-linear unit $σ (Δ ZM P_{[- 1 1]})$ is added to the ZMP regulation controller in this paper, shown in Figure 11. The controller is effective during full contact with the surface phase.

Figure 11.

Zero moment point (ZMP) regulation controller schematic diagram.

$Δ ZM P_{[- 1 1]}$ is an unification in interval [−1, 1], calculated by $Δ ZMP$ .

Δ ZM P_{[- 1 1]} = [\begin{matrix} x_{Δ ZM P_{[- 11]}} \\ y_{Δ ZM P_{[- 11]}} \end{matrix}] = [\begin{matrix} x_{Δ ZMP} / (L_{x} / 2) \\ y_{Δ ZMP} / (L_{y} / 2) \end{matrix}]

(13)

$L_{x}$ and $L_{y}$ are the lengths of the support area in the x- and y-axes. In Figure 7, $L_{x}$ is $FL$ and $L_{y}$ is $FW$ in SSP; $L_{x}$ is $FL + s_{x}$ and $L_{y}$ is $FW + s_{y}$ in DSP.

The $σ (Δ ZM P_{[- 1 1]})$ curve is shown in Figure 12. When $σ (Δ ZM P_{[- 1 1]})$ is in [−d, d], the ZMP regulation controller is not effective. Otherwise, the ZMP regulation controller is effective.

Figure 12.

$σ (Δ ZM P_{[- 1 1]})$ curve.

For the SCUT-I bipedal humanoid robot, d=0.6 is selected. The PD controller is described as follows:

{\begin{matrix} Δ q_{zmpx} = {[Δ q_{zhp} Δ q_{zkp} Δ q_{zap}]}^{T} = {[k_{zp 1} k_{zp 2} k_{zp 3}]}^{T} (K_{Pzp} σ (x_{Δ ZMP [- 1 1]}) + K_{Dzp} \frac{d σ (x_{Δ ZMP [- 1 1]})}{dt}) - {[0 0 1]}^{T} Δ q_{aap 1} \\ k_{zp 1} + k_{zp 2} + k_{zp 3} = 1 \end{matrix}

(14)

{\begin{matrix} Δ q_{zmpy} = {[Δ q_{zhr} Δ q_{zar}]}^{T} = {[k_{zr 1} k_{zr 2}]}^{T} (K_{Pzr} σ (y_{Δ ZMP [- 1 1]}) + K_{Dzr} \frac{d σ (y_{Δ ZMP [- 1 1]})}{dt}) - {[0 1]}^{T} Δ q_{aar 1} \\ k_{zr 1} + k_{zr 2} = 1 \end{matrix}

(15)

$Δ q_{zmpx}$ and $Δ q_{zmpy}$ are the adjusted quantities of $Δ ZM P_{[- 1 1]}$ in the x- and y-axes. $Δ q_{zhp}$ , $Δ q_{zkp}$ and $Δ q_{zap}$ are respectively the hip joint, the knee joint and the ankle joint for the support leg in the pitching direction. $Δ q_{zhr}$ and $Δ q_{zar}$ are respectively the hip joint and the ankle joint for the support leg in the rolling direction. $K_{Pzp}$ and $K_{Pzr}$ are proportional gains. $K_{Dzp}$ and $K_{Dzr}$ are differential gains. $k_{zp 1}$ , $k_{zp 2}$ , $k_{zp 3}$ and $k_{zr 1}$ , $k_{zr 2}$ are weighted parameters. $Δ q_{aap 1}$ and $Δ q_{aar 1}$ are the modifications calculated by the body attitude control in order to offset the attitude controller’s impact on the ZMP controller.

Bipedal robot attitude correction controller

In order to make bipedal robot walk on slopes, an attitude correction controller is designed to control the robot body upright all the time through adjusting the angle of the ankle joint in the rolling and pitching directions. The controlled variable $Δ q_{attitude}$ is shown as follows.

Δ q_{attitude} = [\begin{matrix} Δ q_{aar} \\ Δ q_{aap} \end{matrix}]

(15)

where $Δ q_{aar}$ and $Δ q_{aap}$ are the adjusted quantities in the rolling and pitching directions.

Attitude controller in DSP

The attitude control in the rolling direction is shown in Figure 13(a). The attitude controller superimposes the control input of the PD controller using the torso roll error $ϕ_{b}$ .

Figure 13.

Attitude control explain drawing: (a) attitude control in rolling direction; (b) attitude control in pitching direction.

The equation of rolling direction controller is shown as follows:

Δ q_{aar} = q'_{aar} - q_{aar} = \tan^{- 1} (\frac{{l'}_{rah} - {l'}_{lah}}{l_{aa}})

(16)

l'_{rah} = l_{rah} + K_{Paar} ϕ_{b} + K_{Daar} \frac{d ϕ_{b}}{dt}

l'_{lah} = l_{lah} - K_{Paar} ϕ_{b} - K_{Daar} \frac{d ϕ_{b}}{dt}

where $K_{Paar}$ and $K_{Daar}$ are the proportional and differential gains. $l_{rah}$ , $l_{lah}$ , $l'_{rah}$ and $l'_{lah}$ are distances between the hip joint and the ankle joint.

The attitude control in the pitching direction is shown in Figure 13(b). The attitude controller also superimposes the control input of the PD controller using the torso pitch error $θ_{b}$ .

The equation of pitching direction controller is shown as follows:

Δ q_{aap} = q'_{aap} - q_{aap} = K_{Paap} θ_{b} + K_{Daap} \frac{d θ_{b}}{dt}

(17)

where $K_{Paap}$ and $K_{Daap}$ are the proportional and differential gains.

Attitude controller in SSP

The attitude controller also superimposes the control input of the PD controller using the torso rolling error $ϕ_{b}$ and pitching error $θ_{b}$ .

The equation of the controller in the SSP is shown as follows.

Δ q_{aar} = K'_{Paar} ϕ_{b} + K'_{Daar} \frac{d ϕ_{b}}{dt}

(18)

Δ q_{aap} = K'_{Paap} θ_{b} + K'_{Daap} \frac{d θ_{b}}{dt}

(19)

where $K'_{Paar}$ , $K'_{Daar}$ , $K'_{Paap}$ and $K'_{Daap}$ are the proportional and differential gains.

Experimental simulation and result

Simulation of offline gait planning

According to SCUT-I robot model, a simulation of bipedal humanoid robot walking on slopes is constructed using Matlab 6.5. A slope model is built with a slope angle of 6°.

The parameters are given according to the restriction condition and listed in Table 2. The COG initial co-ordinates are ${[\begin{matrix} 0 & 0 & h_{c} \end{matrix}]}^{T}$ and landing ground point initial co-ordinates are ${[\begin{matrix} 0 & s_{y} / 2 & 0 \end{matrix}]}^{T}$ .

Table 2.

The parameters for walking on slopes.

T _sup	T _db	T _up	T _over	T _down	h _c	Δh_hc	h _foot	s _x	s _y
1.2 s	0.2 s	0.15 s	0.7 s	0.15 s	0.3 m	0.03 m	0.025 m	0.1 m	0.12 m

According to the planning method mentioned in this paper, the robot successfully walked on the slope in simulations, shown in Figure 14(a), and the curves of swinging ankle and hip are shown in Figure 14(b) and (c).

Figure 14.

The simulation of robot walking on slopes by using Matlab 6.5: (a) successfully walking on the slope; (b) the curve of swinging hip; (c) the curve of swinging ankle.

During the walking on slopes process of the robots, the ZMP position can be calculated, and $Δ ZMP$ is the difference between the ZMP and the support area centre. So in the DSP, $Δ ZMP$ should satisfy Equation (20).

{\begin{matrix} - (FL + s_{x}) / 2 \leq x_{Δ ZMP} \leq (FL + s_{x}) / 2 \\ - (FW + s_{y}) / 2 \leq y_{Δ ZMP} \leq (FW + s_{y}) / 2 \end{matrix}

(20)

In the SSP, $Δ ZMP$ should satisfy Equation (21).

{\begin{matrix} - FL / 2 \leq x_{Δ ZMP} \leq FL / 2 \\ - FW / 2 \leq y_{Δ zmp} \leq FW / 2 \end{matrix}

(21)

and FL=0.14 m and FW=0.08 m. So in the DSP, $- 0.12 \leq x_{Δ zmp} \leq 0.12$ , $- 0.1 \leq y_{Δ zmp} \leq 0.1$ ; in the SSP, $- 0.07 \leq x_{Δ zmp} \leq 0.07$ , $- 0.04 \leq y_{Δ zmp} \leq 0.04$ .

During the walking on slopes process of the robots, the curve of $Δ ZMP$ is shown in Figure 15. The DSP curve is shown with a red solid line, the SSP curve is shown with a blue dashed line.

Figure 15.

The curve of $Δ ZMP$ : (a) the curve of $x_{Δ ZMP}$ ; (b) the curve of $y_{Δ ZMP}$ .

After the curve is analysed according to Equations (20) and (21), the offline gait trajectory method mentioned in this paper is proved to be effective in that offline gait planning can make the robot stably walk on the slope.

SCUT-I robot walking on a slope

The SCUT-I bipedal humanoid robot is controlled to walk on a slope whose inclination angle is 6°, as shown in Figure 16.

Figure 16.

SCUT-I robot walking on a slope.

The slope in Figure 16 is supported by two boxes, and the middle of the slope is uneven. So with only an offline gait planning method, it is difficult to make the robot stably walk on the slope.

The online control algorithm that mentioned in this paper is required to be used. By using this method, the SCUT-I robot was controlled to walk on the slope stably and successfully, as shown in Figure 17.

Figure 17.

SCUT-I robot walking on a slope successfully.

The curve of attitude fusion result is shown in Figure 18, where the attitude of robot is controlled at ±7° when the robot walked on the slope. During the process, the attitude correction controller is effective in making the robot hold a certain attitude in order to make walk stably on the slope.

Figure 18.

The curve of attitude result when the robot walked on the slope.

The curve of $Δ ZMP$ is shown in Figure 19, where $Δ ZMP$ is controlled to approach the centre of support area, but at some time, the curve is near to ‘1’ or ‘−1’, because, at that time, the robot’s walking state is in not a full contact phase, the value of $Δ ZMP$ is not accurate and the ZMP regulation controller is not active.

Figure 19.

The curve of $Δ ZMP$ when the robot walked on the slope.

When the robot’s walking state is in a full contact phase, the ZMP regulation controller is effective and controls $Δ ZMP$ to approach [−0.6, 0.6] in order to make robot walk stably.

Conclusions

This paper presents a systematic trajectory generation method for a bipedal robot walking on slopes. The planning of robots walking on slopes is parameterized according to the slope environment. A suitable hip and ankle trajectory are designed based on an IPM. Besides offline gait planning, a landing time controller, landing direction controller, ZMP regulation controller and attitude correction controller are designed to control the SCUT-I robot walking on slopes. During the process, an accurate robot attitude, ZMP position and FSR measurement value are needed. In order to obtain accurate attitude information, an adaptive filtering method is built for the fusion of acceleration and gyroscope sensors based on the Sage–Husa adaptive Kalman filtering method. It is simplified and improved for robot attitude measurement in which the measurement noise covariance $R$ is adaptive. The ZMP position is calculated based on FSR sensors attached to the robot’s feet, and according to FSR sensors, the landing time controller and landing time direction controller are accomplished. At the same time, the walking cycle on slopes is also divided close to the actual slope. Finally, the experiment with the bipedal robot walking on slopes is successful, and the results prove that the proposed walking algorithm is effective and practical.

In the future, some methods can be used to optimize the parameters so that the bipedal robot can walk more stably and more quickly.

Footnotes

Funding

This research is supported by Guangdong Provincial Funds (No. S2011040002784) and Central University of Basic Scientific Research Funds (No. 2011ZM0067 and No. 2011ZM0065). Guangdong Ministry of Education Foundation (No. 2011B090400590).

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