Control of a one-legged robot with energy savings

Control of a One-Legged Robot with Energy Savings

A. Schammass

Dept. of Mechanical Engineering, Universidade de São Paulo

13560-250 São Carlos, SP. Brazil

[e-mail]

G. A. P. Caurin

Universidade Bandeirante de São Paulo

09706-340 São Bernardo do Campo, SP.Brazil

[e-mail]

C. M. O. Valente

Dept. of Mechanical Engineering, Universidade de São Paulo

13560-250 São Carlos, SP. Brazil

[e-mail]

This paper presents a new strategy to control an one-legged robot aiming to reduce the energy expended by the system. To validate this algorithm, a classic method as benchmark was used. This method has been extensively validated by simulations and experimental prototypes in the literature. For simplicity reasons, the work is restricted to the two dimensional case due to simplicity reasons. This new method is compared to the classic one with respect to performance and energy expended by the system. The model consists on a springy leg, a simple body, and an actuated hinge-type hip. The new control strategy is composed of three parts, considering the hopping height, the forward speed, and the body orientation separately. The method exploits the system passive dynamics, defined as non-forced response of the system. In this case, the model is modified adding a spring to the hip. The method defines a desired leg trajectory close to the passive hip swing movement. Simulation results for both methods are analyzed and compared.

Keywords: Mobile robots, walking robots, legged locomotion, passive dynamics

Introduction

Research in legged robots has attracted attention from robotics community because the great potential of these machines to adapt to several terrain types, compared to wheeled robots (Todd, 1985).

Basically, legged robots may be classified in two types and differentiated by how the robots are stabilized (Ringrose, 1997). Statically stable robots move slowly and have a large support, so their dynamics can be ignored. On the other hand, dynamically stable robots are able to achieve higher speeds. Since they have small support, these robots are stable only when they are moving. Most of dynamically stable robots are stabilized with a computer, which calculates how the robot should move.

Although dynamically stable robots can reach high speeds with efficiency, these robots (Raibert, 1986) still have many sensors, actuators and electronic devices dissipating energy (Ahmadi & Buehler, 1997a; Gregorio at al., 1997). For achieving energy autonomy – a requirement for any mobile robot – energy must be saved. A high energy efficiency is specially challenging in legged robots because a large energy exchange is expended by internal movements. Since these movements do not contribute directly for mobility, the energy to produce them must be minimized (Ahmadi & Buehler, 1997a).

Recently, the robotics community has investigated the elastic elements to facilitate the hip oscillation. McGeer (1989) studied an unactuated biped, where both legs were connected to a torsional spring. Raibert & Tompson (1989) showed, using simulations, the passive dynamic running of a simplified one-legged robot with hip compliance. They demonstrated that it is possible to find initial conditions that allow the robot to execute several hops without actuators.

Ahmadi & Buehler (1997b) proposed a control algorithm that stabilizes a simplified one-legged robot with hip spring. This algorithm is based on passive dynamic and aims to reduce the energy expended by the hip. Passive dynamics is defined as the system response without external forces.

This paper presents a control strategy using an one-legged model (Raibert, 1984), with a hip spring. This algorithm is based on passive dynamics and aims at reducing the energy expended by the system. Section 2 presents the model used in simulations. In Section 3, two methods are presented for controlling the system. Section 4 presents the results corresponding to both methods, comparing performance and energy consumption. Finally, Section 5 presents a discussion and conclusions of this work.

Nomenclature

B = ground damping
e = angle error
I = mass moment of inertia with respect to the center of gravity
I' = mass moment of inertia with respect to the hip
K = stiffness
K_p = gain of the angle error
K_v = gain of the angular velocity error
M = mass
T_s = stance time
w = distance from the foot to the hip

x_ERR = linear error of forward speed

x_CG = horizontal projection of the center of gravity

Greek Symbols

DX_s = horizontal distance traveled during stance

c = displacement of actuator

t = torque

h = locomotion time

w = natural frequency

Subscripts

td = relative to touch-down

G relative to ground

L = relative to the leg

H = relative to the hip

e = relative to the effective mass moment of inertia

s = relative to stance time

f = relative to flight time

lo = relative to lift-off

0 = relative to the extremity of the leg (foot)

1 = relative to the center of gravity of the leg

2 = relative to the center of gravity of the body

Model

The model for the study of dynamically stable locomotion is shown in Fig. 1. First proposed by Raibert (1984), the model has only one compliant leg, which is articulated with respect to a body about a hinge-type hip. An actuator is located in this joint, generating a torque t between the leg and the body. The body is represented by a rigid mass, to which the leg is connected. The leg has mass M₁, with mass moment of inertia I₁, and the body has mass M₂, with mass moment of inertia I2. The distance from the lower tip of the leg (the robot’s foot) to the leg center of mass is r₁. The distance from the hip to the body center of mass is.r₂ .For the second control method, presented in Section 3.2, a variation of this model is used, adding a torsional spring with stiffness K_H to the hip.

The total leg length is influenced by a linear spring, a position actuator in series with the spring and a mechanical stop. The leg spring is modeled with one end rigidly connected to the foot and another tip fixed to one side of the actuator. The mechanical stop is modeled as a very stiff spring and a damper.

The position actuator, whose displacement is represented by c, is arranged in series with the leg spring, actuating between the leg and the hip. Changes in actuator length make the energy stored in leg spring to be increased or decreased.

The support surface is modeled as a two-dimensional spring K_G, and a damping B_G. It was assumed that the spring actuates uncoupled in vertical and horizontal directions. The ground compliance and damping influences the robot only when the foot is in contact with the ground, y₀<0. During the flight, the contact forces are zero. The ground stiffness is much larger than the leg stiffness (K_G>>K_L), and the damping coefficient B_G is chosen so that vibrations between the foot and the ground are negligible. This elastic ground model also assumes that the slippage never happens.

Control Strategy

The control strategy is divided in three parts, assuming a weak coupling among the movements. It treats the vertical movement, the forward speed, and the body orientation as three separately problems. The first part of the strategy excites the vertical movement of robot and regulates its height thrusting the leg actuator at each hop. The second part stabilizes the forward speed, placing the foot (contact point) at a position that it will give the required acceleration for the phase when the robot is in contact with the ground (stance). The third part maintains the body upright, actuating in the hip during stance. The three parts of the system are synchronized in function of robot’s events.

The vertical control is based on compensating the energy losses. The energy difference required for achieving the desired height is estimated. Then, the difference is converted to terms of actuator displacement. The actuator movements are synchronized by the identification of events in hopping cycle. The actuator lengths in the maximum spring compression and shortens just after the instant that robot leaves the ground (lift-off).

The sections 3.1 and 3.2 present two methods for controlling the speed and orientation. The first method is based on strategy presented by Raibert (1986). This method places the foot during flight to control speed and actuates the body angle during stance to control orientation. The second method performs the same task, but the trajectories for foot placement are generated in order to minimize the energy consumption. Simulations were realized for both algorithms and the results are presented in section 4.

Method 1: Foot Placement and Orientation Control

First, the speed control is calculated by finding the nominal movement that keeps the forward speed constant and the body balanced. Then, this movement is modified to eliminate the deviations. The algorithm consists of two parts: the first part uses foot placement to control the forward speed, and the second uses hip movement in order to control the orientation during stance.

During stance, the leg moves backwards with respect to the body. This motion is symmetric about the point localized in the middle of stance (neutral point). This symmetric motion causes neither momentum, nor acceleration because the center of gravity spends approximately the same time in front of neutral point and behind it. Therefore, the average of momentum and acceleration are zero throughout stance.

When the forward speed deviates from its desired value, the foot is positioned with respect to the neutral point correcting the error (Fig. 2). If the foot is placed to the neutral point (Fig. 2b), the body travel throughout a symmetric trajectory, what leaves the robot with no forward acceleration. When the foot is placed before the neutral point (Fig. 2a), the body is accelerated during stance. When the foot is placed after the neutral point (Fig. 2c), the body is decelerated during stance. A linear error of forward speed determines the distance that the foot should be positioned with respect to the neutral point.

The diagram of Fig. 3 illustrates how the method works. To compute the neutral point, half of horizontal distance traveled during stance (Equation 1) is added to the horizontal projection of the center of gravity (Equation 2). If the stance time Ts and the forward speed are known, the horizontal distance traveled during the stance is given by:

and the horizontal projection of the center of gravity is expressed by:

Then, the linear error of forward speed (Equation 3) is added to Equation 4. The following analysis is realized in a coordinate system that moves with hip. The linear error of forward speed gives the corrective feedback given by:

where is the desired value for the speed , and K₁ is the feedback gain. Then, the horizontal foot position when it touches the ground (touch-down) is given by:

Considering the kinematics of model, the leg angle q₁ in touch-down is obtained:

For tracking the desired angle, it is used a linear controller given by Equation 6.

where e_q1= q_1,td – q₁ is the leg angle error, and K_pf, K_vf are the feedback gains during flight. A similar linear controller is used for orientation control during stance (Equation 7). These gains are calculated by simplifying the model to a linear system, which takes into account the body inertia and neglects the leg coupling. Since the closed loop system is obtained, it is possible to choose appropriate gain values to obtain the desired performance. The performance parameters are chosen so that the leg angle regulation is faster than the flight time.

where e_q2= q_2,d – q₂ is the body angle error, q_2,d is the desired value, and K_ps, K_vs are the feedback gains during stance. As in the flight time, these gains are also calculated using the simplified linear system. In this case, the performance parameters are chosen so that the body angle regulation is faster than the stance time.

Method 2: Foot Placement and Orientation with Energy Savings

The second method as illustrated in Fig. 4, it is an extension of the first one and it aims at reducing the energy expended by the hip actuator. For achieving this, a torsional spring with stiffness KH is added to the hip. The idea of this strategy is based on hip passive movement, as illustrated in Fig. 3. The natural frequency of hip oscillation is approximately given by:

where I_e = I'₁I'₂/(I'₁+I'₂) is the effective mass moment of inertia, and the terms I'₁ = I₁+M₁r₁² and I'₂ = I₂+M₂r₂² are the mass moment of inertia with respect to the hip. Considering the flight time approximately equal to the stance time, the spring constant is chosen so that the period of oscillation is equal to step time.

As illustrated by Fig. 5, since the leg has to hit the ground with desired angle q_1,td, the hip oscillation must be synchronized with vertical movement. To obtain such synchronization, time is not a suitable parameter because flight and stance times are subject to variations during locomotion. Therefore, it is necessary to use a variable named locomotion time (Ahmadi & Buehler, 1997b), which describes the vertical motion independent of the operation conditions (e.g. hopping height).

The locomotion time h is defined as a scalar function, which maps the flight phase in a fixed interval [0 1], from the lift-off (h_1o= 0) to the touch-down (h_td= 1).

Therefore, the locomotion time is defined during flight:

where is the velocity of the body just after lift-off. Using locomotion time and touch-down angle calculated in the previous method (Equation 5), the desired leg angle trajectory during flight is defined:

To track this trajectory, a controller based on simplified model inverse dynamics is used. This type of controller is commonly used in manipulators (Koivo, 1989).

where e_q1= q_1,d – q₁ is the leg angle error, and K_pf, K_vf are the feedback gains during flight phase. The gains are chosen such way the controller response is faster than the desired trajectory. Orientation control uses the same strategy as method 1. However, since the system has hip spring, a controller based on the inverse dynamics of the system is used.

where e_q2= q_2,d – q₂ is the body angle error, and K_ps, K_vs are the feedback gains during the stance phase.

Results

In this section, the results of both methods are presented. The results of performance and energy consumption are compared and analyzed.

Fig. 6a shows the results for the first test using method 1. This test evaluates the system capability of achieving a desired speed (= 1m/s). The model starts hopping in place (), and then is accelerated to 1 m/s. The data from this figure demonstrates the robot reaches the steady state about 3.5 seconds. During steady state, the speed varies in the range 0.76 to 1.23 m/s.

Fig. 6b shows the results of the same test using the second method. The data of this figure demonstrates a similar performance as method 1. The system reaches the steady state about 3.5 seconds. In this case, the speed varies in the range 0.93 to 1.35 m/s during the steady state. In both methods, it was possible to control the forward speed up to 2 m/s.

Fig. 7 shows the energy expended in 10 seconds by the hip actuator, using both methods presented in Section 3. The results demonstrate that the second method obtained approximately up to 67% of energy savings with respect to method 1.

Conclusion

In this paper, a new strategy was presented for controlling an one-legged robot. This strategy is based on the classic method developed by Raibert (1986). The major contribution of this strategy lies on the improvement of the classic one with respect to energy consumption. For evaluating the new method, both classic and new strategies are simulated and their performance and energy consumption are compared.

In both methods the control is divided in three parts: Hopping height control, forward speed control, and orientation control. The classic strategy (method 1), proposed by Raibert (1986), uses the foot placement during flight phase for controlling the speed. During stance, this algorithm controls the body orientation.

The new strategy (method 2) uses a model with hip spring. This method is an extension of first method using the foot placement during flight, and orientation control during stance. In this strategy, the foot placement is adapted to reduce the energy consumption. To achieve result, the desired leg angle trajectory is based on hip passive behavior.

The results demonstrate that both performances are similar. Both methods have similar response time and steady state variation. Nevertheless, the new method has shown considerable energy savings.

Manuscript received: February 2000. Technical Editor: Átila P. S. Freire.

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Publication Dates

• Publication in this collection
  04 Oct 2001
• Date of issue
  2001

History

• Received
  Feb 2000