Zhuo Ge, Qingsheng Luo, Baoling Han, Qi Na and Huashi Li(1.School of Mechatronical Engineering, Beijing Institute of Technology, Beijing 10008 China;2.Marine Design and Research Institute of China, Shanghai 20001 China;3.Science and Technology on Water Jet Propulsion Laboratory, Shanghai 20001 China;.School of Mechanical Engineering, Beijing Institute of Technology, Beijing 10008 China)

Legged animals present an efficient and harmonious locomotion ability, capable of walking and running on unstructured terrains. Hence, the development of bio-inspired robotic controllers seems to be a feasible and robust way to obtain an efficient and robust robotic locomotion, mimicking their biological equivalents. Inspired by legged animals, quadruped robots can step over obstacles and walk through irregular terrains. They are stable and powerful than biped robots, which make them well suited for outdoor tasks.

It is known that locomotion in animals is generated at the spinal cord by a combination of central pattern generators (CPGs)[1]. CPGs can be simply defined as biological neural networks capable of producing rhythmic patterns in absence of sensory information. When the CPG model is applied to walking gaits, the outputs of the CPG are changed to positions and angles which inspire the musculoskeletal system. Moreover, the sustained locomotion movement in the presence of environment changes is very likely related to the important role of sensory feedback in shaping the CPG’s activity. Therefore, the outputs of CPG should correspond to biological signal feedbacks[2].

Fukuoka et al.[3]constructed a CPG control network for their quadruped robot Tekken by using improved Matsuoka neuron oscillators. Based on experimental results Tekken achieved adaptive dynamic walking considering sensory inputs which change the period of original active phase of the CPG model. Cheetah-cub, a fast trotting quadruped robot developed by EPFL[4-5], have an open-loop locomotion controller on the basis of CPG network in an open loop mode. The oscillator outputs were applied as position signals at the robot’s active joints. Arena et al.[6-8]proposed a CPG-based control strategy for hexapod walking robot combining with cellular non-linear network (CNN). The hierarchical architecture of CNN-CPG structure can be used to modify the speed of the robot that can be controlled to follow a reference speed signal. Zhang Xiuli et al.[9-10]presented modified CPG networks based on Kimura’s CPG model to control the quadruped robot Biosbot’s hip joints and introduced a hip-to-knee mapping function outside the CPG network to control the robot’s knee joints by constructing the vestibular reflex model and flexion reflex model. Hence the Biosbot can achieve the action of up and down slopes, surmounting obstacles and stabilizating its posture. However, the core CPG model have a poor integration with the reflections.

Therefore, it is important to study the CPG corresponding to biological signal feedbacks and establish the method to realize the mapping from the CPG to the musculoskeletal system in the fields of robot motor control, especially the slope terrain locomotion control. In this paper, a bionic modified vestibular reflection model is applied to the CPG rhythm motion control model of a quadruped robot with a particular configuration of inward-knee joints. While walking on slopes, the variation of robot’s trunk is introduced into the presented CPG model as a feedback term. With the feedback term, the fusion of the vestibular reflection model and the CPG model is realized. This modified vestibular reflex model can reduce the fluctuation of center of gravity (COG) and improve the locomotion stability effectively.

1 Quadruped Robot and CPG Model

As shown in Fig.1, a bionic hydraulic quadruped robot is developed as an autonomous mobile platform adapting to the irregular terrain. The quadruped robot consists of one trunk, two pairs of side-sway parts, two pairs of thigh parts and two pairs of shank parts. And the forelegs and hind legs are symmetrically distributed. The overall dimensions of the robot are 1 500 mm×500 mm×1 200 mm and the total weight is 120 kg. Each leg has two joints (hip joint and knee joint) driven by hydraulic cylinders and three degrees of freedom (DOFs): hip rolling, hip pitching and knee pitching.Focusing on analyzing the basic characteristics of the quadruped robot movement in the leg-sagittal plane, each leg of the virtual prototype model has two rotary motions: hip pitching and knee pitching.

1.1 Biological CPG control model

CPGs are neuron circuits that can produce coordinated oscillatory signals. Moreover, the sensory reflexes and higher-level stimulator can modulate the activity of CPGs[11]. Inspired by the biological feature of CPG, a biomimetic CPG control network is proposed,which is modulated by vestibular reflection feedback. The Hopf oscillator as a simple harmonic oscillatoris selected as the CPG unit model and the amplitude and frequency of the output signal are easy to control[12]. Each joint is controlled by one oscillator, and all the oscillators build up the CPG network to fulfill particular modes of locomotion.

While the traditional Hopf oscillator can generate stable and periodic oscillation signal, but the ascending and descending parts of the output curve have equal durations. In order to achieve an independent control of the durations of these two parts, the modified Hopf oscillator can be described as:

(1)

wherexandyare state variables of the oscillators,r2=x2+y2,μ>0 determine the amplitude of oscillations. Positive coefficientsεis correlated with the speed that the oscillation converges to a limit ring. Andωis the frequency of the oscillations.ωswis the frequency of swing stance,ωstis the frequency of stance phase,ais a positive constant which controls the speed ofωtransforming formωswtoωst.βis the duty factor which is the proportion of stance phase to the gait period.

The virtual prototype model of the robot has 8 active connections among 4 hip joints and 4 knee joints as mentioned before. If all the 8 oscillators are directly connected, the CPG control network will be complicated and inefficient. With the 8 Hopf oscillators are divided into 2 layers, one layered CPG control network model is constructed, as shown in Fig.2.

Fig.2 CPG control network of the hydraulic quadruped robot

Four coupled oscillators controlling 4 hip joints comprise the first layer using a symmetrical mesh topological structure. In the second layer, each knee joint oscillator couples with its homologous hip joint oscillator unidirectionally. Then outputs of this network are used as joint angle control signals. The angle control signals of hip joints are obtained from the coupled 4 hip joint oscillators; the control signal of each knee joint is generated by the knee joint oscillator unidirectional coupled with the hip joint in the same limb.The mathematics model of presented CPG control network is

(2)

(3)

1.2 Rhythmic trot motion by CPG network

The movement laws between the hip joint and the knee joint within the same limb of a tetrapod mammal were summarized as follows[13]:

① There is a fixed phase relationship between the hip joint and the knee joint during normal walking movement;

② The hip and knee joints are synchronous during swing phase;

③ The hip joint pulls backwards while the knee joint is basically hold still during a stance phase.

Inspired by these movement laws, the motion relationship between hip joint and knee joint within a limb can be expressed in Fig.3. As it shown,y<0 of the hip joint control signalygenerated by the presented modified Hopf oscillator corresponds to the rising of output knee joint control signalx. This correspondence is conformed to the phase relationship of motion between the hip and knee joints.

Fig.3 Movement relation between hip and knee joints

As mentioned before, K is the connection-weight matrix of the CPG control network. It determines the output pattern of the CPG control network which defines different gaits.Quadruped animals have different gaits to fit for different terrains and speeds. Trot is a common gait in robot walking control, in which the diagonal legs lift simultaneously while the other two supporting the body. In this paper, we focus on the control of the trotting gait of a quadruped robot on slope terrain. There, for the trot gait, the duty factorβ=0.5, henceωst=ωsw, the matrix K as

(4)

In Fig.4, a relative phase of trot gait is described. LF, RF, RH and LH represent the left front leg, the right front leg, the right hind leg, and the left hind leg respectively.φiis the phase ofith oscillator.

In order to calculate the parameters of the gait control model, the kinematic diagram of one limb in a trot gait period is shown in Fig.5.

The joint balance angleθ0in the joint coordinate system is defined as the angle between the limbs and the vertical direction. In the case of static stability of the robot, based on the principle of simplification, both the balance angles of the hip and knee joints areθ0=30°. For the trot gait, the duty factorβ=0.5, during the support phase and swing phase, the body will move forwards/2 respectively.sis the length of one gait,vis the velocity of the robot,Tis the period of the walking cycle,lis the length of one segment of the limb. The swing amplitude of the hip joint and the knee jointAhandAkcan be calculated by the other parameters.

Fig.4 Relative phase of trot gait

Fig.5 Kinematic diagram of the single leg movement in trot gait mode

2 Biological Reflex CPG Model

2.1 Vestibular reflex CPG model for slope locomotion

CPG network is the center of the animal’s motion control neural network which can either receive control commands from the advanced central nerve system or respond to feedbacks from various receptors in the body. The feedback signal can adjust the output of the CPG to change the characteristics of the CPG control network, thereby changing the rhythm motion mode (amplitude, frequency, phase, etc.), so that the movement can be changed according to changes in the external environment. Hence the feedback signals are crucial for the environmental adaptive locomotion of the quadruped robot. Abstracted from the neural reflexes of legged animals, a vestibular reflex sensory model for control of trotting on slope is employed to the presented CPG controller. By introducing the feedback signals feedxiand feedyi, the vestibular reflex CPG network model could be described as:

(5)

By studying the slope movement of legged animals, it can be found that when the animal moves on slope, the body spine line is not parallel to the slope, nor parallel to the horizontal plane. It maintains a certain angle with the slope. By reducing the absolute angle between the body and the horizontal plane, the position of the center of gravity projected on the supporting surface is changed to ensure the balance of the body.

The attitude adjustment of the body can be achieved by changing the equilibrium position or range of motion of the front and hind legs that contracting front legs while stretching the hind legs when trotting on uphill slopes and stretching front legs while contracting the hind legs when trotting on downhill slopes.

Fig.6 demonstrates the relationship between the change of equilibrium position of the joints Δθand body posture angle Δαwhen the robot is trotting on the uphill slope.l0is the length of robot’s body,lis the length of one segment of the limb, andθ0is defined as the original joint balance angle. The adjustments of joint equilibrium position are listed in Tab.1.

Summarized by Ref.[14], the sufficient condition for the quadruped robot to realize stable slope movement and maintain the optimal energy consumption is that the attitude angle of the robot’s body is approximately linear with the slope as

Δα=q1α

(6)

whereq1=0.24[14].The body posture angle Δα

(7)

whereαis the slope gradient,Lis the length between the homolateral hip joints,lis the length of one segment of the limb, andθ0is the original joint balance angle when the robot walking on flat ground.

Fig.6 Relationship between the change of equilibrium position and body posture angle

Tab.1 Adjustments of joint equilibrium position for slope locomotion

For a CPG model with a specific set of parameters, the amount of change in the equilibrium position cannot exceed the amplitude of the CPG output signal, otherwise the output signal will be completely in the positive or negative axis, as a result the CPG’s oscillation function will be incapacitated. Considering Δθ∈[0,0.1 rad], we can calculate the approximate relationship between Δθand Δα:

Δθ=q2Δα

(8)

whereq2=2.26, obtained by the fitting curve of Eq.(7).

The magnitude of the feedback signal is equal to the amount of change in the intermediate value of the CPG output signal and also equal to the amount of change in the joint equilibrium position.

|feed|=Δθ=q2q1|α|

(9)

Combining with the different movement of the knee-type joint and elbow-type joint, the vestibular reflex modelof the all inward-knee joints configuration of the robot can be expressed as

(10)

Noticing that the adjustment of the knee position is opposite to the hip joint in the same limb, so the feedback signal of the knee joint is opposite to feedback of the hip joint.

2.2 Modified vestibular reflex model

The quadruped robot we presented has the joint configuration of all inward-knee joints that the forelegs are elbow-type joints and the hind legs are knee-type joints. With the vestibular reflex model proposed in Eq.(10), the body posture angle can be kept basically stable during the movement of the robot, and satisfies the sufficient condition of the slope locomotion.

However, the discrepancy between vertical distances from the hip joint to the slope when the hip joint reach front limit position and posterior limit position deduced by the ordinary vestibular reflex feedback model could cause the COG of the robot moves downwards during the next support phase. Therefore, in the duration of the uphill or downhill slope movement, the COG of the trunk will experience obvious unstable fluctuation.

Based on the above analysis, we use a modified algorithm to calculate the feedback parameters. Ensuring the auxiliary line connecting the hip joint in the equilibrium position and the foot end is perpendicular to the slope, while the knee joint equilibrium position is kept constant, the front limit position and posterior limit position of the hip joint are adjusted. Then we can calculate the modified equilibrium position of the hip joint which will keep the COG of the robot’s body ascending or descending continuously, as shown in Fig.7. The adjustments of modified joint equilibrium position are listed in Tab.2.

Fig.7 Kinematic diagram of modified joint equilibrium position for slope movement

Tab.2 Adjustments of modified joint equilibrium position for slope locomotion

Hence， the modified vestibular reflex feedback model and the hip and knee angle control signals can be expressed as

(11)

(12)

3 Co-simulation of the Slope Locomotion

3.1 Simulink model of the reflex CPG network

In order to verify the effectiveness of the proposed CPG control scheme, gait simulations of quadruped robot are carried out by the co-simulation of ADAMS and MTLAB/Simulink. According to the mathematical model of the vestibular reflex CPG network, the corresponding control system model is established by MTLAB/Simulink.Consisting of an angle variation distribution module and an angle adjustment calculation module of the equilibrium position, the vestibular reflex model is integrated into the original CPG network,as shown in Fig.8 and Fig.9.

Fig.8 Simulink model of the vestibular reflex model

3.2 Co-simulations

The dimension parameters of the quadruped robot and other simulation parameters used for the simulation can be found in Tab.3.

To evaluate the performance of the proposed vestibular reflex CPG control model, several systematic analyses were carried on. First of all, uphill locomotion simulation experiments of the non-reflex, the vestibular reflex and the modified vestibular reflex CPG control model were conducted. The COG displacements of the robot in the direction of forward movement are shown in Fig.10.

From the simulation results, it can be observed that without the vestibular reflection feedback, the robot suffered severely slipping which causes the minimum foreword movement and suffered COG fluctuation. Controlled by the normal vestibular reflex CPG model, the robot is able to climb up the slope and eliminate the slipping effect appropriately. However, fluctuation of COG is still obvious. Maximum movement distance and nearly no COG fluctuation showed that driven by the modified vestibular reflex CPG model the quadruped robot can trot on slope efficiently and stably, and can adjust its posture effectively based on the feedback signal.

Fig.9 Simulink fusion model of the vestibular reflex model and the CPG network

Tab.3 Simulation parameters

Fig.10 Simulation of COG displacement in different robot models (x axis)

3.3 Adaptive trotting by vestibular reflex model

An adaptive trotting co-simulation experiment is designed for the quadruped robot to validate the vestibular reflex model’s ability of adapting to the environment. In this experiment, the quadruped robot is supposed to walk uphill from the plain ground and change its posture as needed. When trotting on the plain ground, there is no feedback signals so the robot can trot driven by the normal CPG network. And when trotting uphill, as variations of pitch angles are detected by the acceleration sensor mounted in the body, the slope angle will be introduced into the vestibular reflex CPG model (modified reflex model if not stressed) as a feedback signal. The output curves of the front and hind hips move gradually downward, that is, their equilibrium positions decrease continuously, and the output curves of the knees move against the hind hips correspondingly. Thereby, the quadruped robot transform its plain locomotion posture to the slope locomotion posture to maintain its balance and counteract slipping. Fig.11 show the screen shots from the simulation, the corresponding COG displacements trajectories, and the diagram of body posture.

Fig.11 Simulation of adaptive trotting by vestibular reflex model

The results show that after the front legs step on the slope terrain, the body posture gradually altered in reference to the slope degree. Influenced by the modification of output hip and knee joint angle control curves, mutations of the ground-reaction force occur after the adjustment of posture. Then the robot can maintain steady locomotion onslopes.

4 Conclusions

In this paper, a two-level CPG control network is constructed based on modified Hopf oscillators, which imitated animals’ control mechanism. A biomimetic vestibular reflex feedback model is presented with the inspiration from the neuronal principles underlying the locomotion of tetrapods. The vestibular reflex model is modified according to the particular configuration of all inward-knee joints for the presented quadruped robot and the trot gait pattern, which could stabilize the COG fluctuation of the robot’s body and countervail slipping. Focusing on slop locomotion of the quadruped robot with trot gaits, the presented control system is validated in the simulation.

Results of the contrast experiments and adaptive movement co-simulation have shown that driven by the presented modified vestibular reflex CPG model, the quadruped robot is able to have stable dynamic trotting on slope terrain and have terrain adaptive ability in complex situations. Future work will involve experiments on real robots and imitate more biologic reflex actions, which can help the robot to adapt to more complex and unpredictable environments.

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