Humanoid Robot Gait Planning and Experimental Verification Lei Zhang*1, Xiaokai Feng2, Zhe Qiu3 Department of Engineering, Ocean University of China, China zhanglei1107@ouc.edu.cn; 2fengxiaokai2006@126.com; 3qz93396@163.com
*1
Abstract This paper presents an optimized method of gait planning for a small biped humanoid robot developed independently. The three-dimensional inverted pendulum model is adopted to gain the information of the center of gravity (COG) trajectory and angle of each robotic joint to meet the ZMP stability, transform the information into angle and speed of steering, and then realize the stable walking of the robot. The dynamic biped walking experiment based on walking parameters was taken to verify the feasibility and validity of the gait planning method. Keywords Biped Robot; Gait Planning; Three-Dimensional Inverted Pendulum; Inverse Kinematics
Introduction With the development of robotics technology, stable walking theory has been remarkably development. The mainly used gait planning methods are based on bionic kinematics, model and intelligent algorithm[1]. Based on these theories, researchers have developed numerous prototypes realizing complex tasks, such as walking on flat, up and down stairs, fall and rise[2][3]. However the experimental results are not ideal due to the complexity of control, poor stability and strict limitations of the environment. This paper presents a simple and practical method of gait planning, which is suitable for humanoid robots with the same structure of legs. We adopt the characteristics of parameter of the developed small biped humanoid robot, and validate our proposed method by dynamic biped walking task.
FIGURE 1. THE ROBOTIC PROTOTYPE
Robotic Model The robotic prototype is based on the principle of bionics, which parameters have the same proportion as human body. We try to reduce the height of COG and increase the area of feet to improve the walking stability effectively. As shown in Figure 1, the DOF of the prototype is distributed as follow: 12 DOF for legs, 6 DOF for arms, 2 DOF International Journal of Automation and Control Engineering, Vol. 4, No. 1—April 2015 2325-7407/15/01 051-7 Š 2015 DEStech Publications, Inc. doi:10.12783/ijace.2015.0401.12
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Lei Zhang, Xiaokai Feng, Zhe Qiu
for head[4]. In addition, we define two virtual joints to facilitate our research. Digital servo AX-12A is adopted as the joint driver, which has high control accuracy and real-time status information feedback. The world coordinate system is established as shown in Figure 1, which defined the projection point of the COG on the ground as the origin of the coordinate, the orientation of x-axis points to the front of the robot. The origin of the local coordinate system is the projection point of ankle joint on the ground, which axes are parallel to the world coordinate system. Design of Dynamic Walking The difficulty of biped robotic control is dynamic walking. We treat the robot legs as massless rods to support the COG movement. The robot is mostly supported on a single leg during the walking cycle, thus we simplify the robot as a three-dimensional inverted pendulum model[6][7][8], which can concentrate the quality of the entire robot to the center of mass, and simplify the supporting legs as scalable and massless struts. In this paper, we get stable trajectory of COG by adjusting the position of foothold according to the given movement, and then generate pose information of each joint based on planned swung legs calculated by inverse kinematics, finally achieve stable biped walking. We decompose the three-dimensional motion into two-dimensional plane movement of XOZ and YOZ, in order to simplify the problem. Design of Walking Parameters As shown in Figure 2, we set the robotic step length as s, step width as d, P0 ~ P5 as the footholds.
FIGURE 2. SCHEMATIC DIAGRAM OF THE ROBOT WALKING
FIGURE 3. INVERTED PENDULUM MODEL
Determine the Walk Cycle The walk cycle of gait planning is an important indicator of the performance of dynamic walking and is useful for the subsequent analysis. Here, we use the linear inverted pendulum model to determine the walk cycle, as shown in Figure3. We set the support point as the origin of coordinate, the movement of the robotic center of mass as the direction of x-axis. Then the kinematics relation between the robot center of mass acceleration and position is described as: x = ( g / z) x
(1)
In eqn. (1), x and z are the coordinates of COG location, g is the gravitational acceleration[9]. When z is a constant, we can get: = x(t ) x(0) cosh(t / TC ) + TC x (0) sinh(t / TC ) t ) x(0) / TC sinh(t / TC ) + x (0) cosh(t / TC ) = x( TC ≡ z / g
(2) (3) (4)
The COG position x(0) and speed x (0) are set as the initial conditions, TC is a constant related to the height of COG
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and gravitational acceleration, x(t ) and x (t ) are position and speed of COG, respectively. Define the orbital energy of robot COG is a constant value E, transformed eqn. (1) and multiplied by x , then integrate over x, we can get: 1 2 g 2 x − x = E 2 2z
(5)
LHS and RHS represent kinetic energy, potential energy, respectively. According to the definition of the orbital energy, before switch of supported foot, g 1 − x 2f + v 2f E= 2z 2
(6)
In eqn. (6), x f is the horizontal position of COG which related to the previous foothold, v f is the speed of COG when supported foot is switching, which is shown in Figure 4. The basic condition for the switch of supported foot s g 1 g 1 is v f ≥ 0 . According to the law of conservation of energy, we can get − x02 + v02 = − x 2f + v 2f , x0 = − , v0 is 2 2z 2 2z 2 the initial speed, s is the step length. The condition for the switch of supported foot is: v f ≥ 0 , when x f = 0 .Solve it, then v0 ≥
s 2
g z
FIGURE 4. INSTANTANEOUS STATE WHEN SWITCH THE SUPPORTED FOOT
Setting the supported time of the supported foot is Ts. The trajectory of robotic COG is a hyperbola which is symmetrical about the y-axis during the period of [0 Ts]. According to its symmetry, the termination conditions will be (− x0 , v0 ) if the initial condition is along x-axis, substitute them into eqn. (2), the solution is (7)
− x0 = x0 C + TC v0 S
In eqn. (7), TC ≡ z / g , C ≡ cosh(TS / TC ), S ≡ sinh(TS / TC ) , then we can get, v0 = −
C +1 x0 TC S
(8)
Then we can obtain the mapping between initial speed and walk cycle according to the different step length and the given COG height. The initial speed can be obtained, from the given step length and walk cycle. We can get the minimum speed by eqn. (8) depended on the parameters of COG height and step length, and then we can calculate the longest movement cycle T by the transition time formula:
τ = TC ln
x0 − TC x0 x1 − TC x1
or τ = TC ln
x1 + TC x1 x0 + TC x0
,
x1 , x1 are the position and speed of COG at time , respectively. Note that the minimum speed is approximated in
order to avoid singular. Theoretically, the robotic walk cycle can be chosen from (0 T). The actual walk cycle is set as 0.4s on the consideration of control accuracy and performance parameters of prototype.
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Lei Zhang, Xiaokai Feng, Zhe Qiu
Calculate the Foothold We can get the orbital energy along the x-axis at the beginning of walk with given initial conditions ( x0 , v0 ) according to the definition of the orbital energy. g 1 − x02 + v02 E= 2z 2
The relationship between position and speed during walking process is: 1 g − x2 + v2 E= 2z 2
As shown in Figure 5, line 4 shows that the COG with low initial speed could not pass through the supported foot and the robot would fall backwards due to gravity when the COG speed is zero. Line 1 shows that the COG could pass through the supported foot with a high initial speed, thus the speed of COG would continuous be high when the swing foot put down, which could result in poor stability. Line 2 shows the optimal motion curve. Line 3 shows the critical result when COG passes through the supported foot. When the walk cycle and the expected foothold are determined, the selection of switching time of supported foot directly affects the walking speed and stability of the next step. As shown in Figure 6, the initial speed of COG of the next step will be lower if the switching is put forward, which would result in fail of the walking process; otherwise, the walking system may be unstable and is not capable to generate subsequent path planning.
FIGURE 5. THE RELATIONSHIP BETWEEN POSITION AND SPEED OF GOG
FIGURE 6. EFFECT ON FOOTHOLD SELECTION OF NEXT STEP
We can have a successful walking process by adjusting the position of foothold. In order to choose the foothold properly, an error evaluation function is defined: N ≡ a ( x d − x (fn ) ) 2 + b( x d − x (fn ) )
x d , x d are the expected foothold position and speed, respectively. x (fn ) , x (fn ) are the final conditions of the nth step, a and b are positive weighted factors. According to
∂N = 0 , we can get the foothold P that can make error evaluation function minimal. ∂P
The Trajectory of Ankle Joint and COG From the analytical function of the robotic COG, we can obtain the COG trajectory by superposing the trajectory of x-axis and y-axis. We choose the first-order sine function as the trajectory of the swing leg depended on the human walking process.
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f ( x ) = Asin (ω x )
A denotes the height of swing foot. ω = 2π / L , L means the step length, x ∈ [0, L / 2] . Determination of Angel of Each Joint According to postures of robot body and foot, we get the angel of each joint by analytical method.
FIGURE 7. DIAGRAM FOR INVERSE KINEMATICS
The steps of inverse kinematics calculation method are as follow: 1. Get positions and attitudes of COG and ankle joint in the world coordinate system. 2. Get the distance D between COG and hip joint, the length A of thigh and the length B of shank. 3. Calculate the position of hip joint P2 in the world coordinate system. 4. Calculate the position vector r of hip joint in the ankle joint coordinate system. 5. Calculate the distance C between hip joint and ankle joint. 6. Calculate the angle q5 of knee joint according to A, B and C. 7. Calculate the angle of line B and C. 8. Calculate the rolling and pitch angles of ankle joint according to the vector r. 9. Calculate the attitude information of hip joint. We can get the angle of each joint of the robot with available attitude of ankle joint and COG basis on the inverse kinematics. Experimental Verification The main performance parameters of the prototype are shown as follow: Name Height Height of COG Height of hip joint Height of ankle joint Foot width Step width
Parameter 450mm 200mm 219mm 33mm 62mm 74mm
Name Weight Degrees of freedom Height of knee joint Leg length Foot length Step length
Parameter 1.8kg 20 126mm 186mm 104mm 30mm
The parameter of step length is set according to the same proportional relationship from human body and walking process. The height of COG is a set value considering for the case of slight bending of two legs, which ranges from 100mm to 210mm. The walking stability will be increased by reducing the height of the COG appropriately during walking process, which would not affect the experimental results. According to eqn. (8), we can get the relationship between initial speed and walk cycle, as shown in Figure 8, the initial speed has a limited minimum when the walk cycle tends to infinity. Based on different centriod heights, as shown in Figure 9, we can get the various curves of initial speed when the
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walk cycle changes. The height of centriod may be set arbitrarily within the effective range, which does not affect the motive trend of prototype.
FIGURE 8. THE RELATIONSHIP BETWEEN INITIAL SPEED AND WALK CYCLE BASED ON VARIOUS STEP LENGTHS
FIGURE 9. THE RELATIONSHIP BETWEEN INITIAL SPEED AND WALK CYCLE BASED ON DIFFERENT HEIGHT OF COG
FIGURE 10. THE PROJECTION POINT OF COG TRAJECTORY
FIGURE 11. THE WALKING PROCESS OF THE PROTOTYPE
The COG trajectory of the robot relied on the proposed gait planning method is shown in Figure 10. The walking process of the prototype is shown in Figure 11. We can figure out that the proposed gait planning method is effective. Conclusions The proposed gait planning method is efficient for off-line planning of biped walking robot, which provides basis method and stable condition for robot dynamic walking process. Moreover, it is beneficial to modify the mechanical structure of robot depended the analysis of robot basis parameters according to the proposed method. Furthermore, we can achieve a stable walking process with the information of gestures, pressure sensors and online modification. This method can also be applied to other biped robot with similar leg structure. REFERENCES
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Yu Zhiwei. “Study on humanoid gait planning and stability of biped robot” PhD diss., Harbin Engineering University, 2008.
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Zhe Qiu, Lei Zhang, Yang Tian, Xiaokai Feng, Shengyuan Zhang. “An Optimized Design of Humanoid Robot Distributed Control System”. Applied Mechanics and Materials, 2014, 541: 1043-1048.
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Lei Zhang (1974- ) received the Ph.D. degree in intelligent control direction of Production Systems Engineering from Ibaraki University of Japan in 2006. He is currently an associate professor in Ocean University of China. His research interests include intelligent control and intelligent robot. Xiaokai Feng(1988- ) graduated from OUC in 2010,bestowed Bachelor degree. He is a post graduate student in OUC, his research interest is biped robot. Zhe Qiu(1989-) recived the MS dgree from OUC in 2014. His research interest is biped robot.