Abstract

In recent years, there has been increasing interest in the study of gait patterns in both animals and robots, because it allows us to systematically investigate the underlying mechanisms of energetics, dexterity, and autonomy of adaptive systems. In particular, for morphological computation research, the control of dynamic legged robots and their gait transitions provides additional insights into the guiding principles from a synthetic viewpoint for the emergence of sensible self-organizing behaviors in more-degrees-of-freedom systems. This article presents a novel approach to the study of gait patterns, which makes use of the intrinsic mechanical dynamics of robotic systems. Each of the robots consists of a U-shaped elastic beam and exploits free vibration to generate different locomotion patterns. We developed a simplified physics model of these robots, and through experiments in simulation and real-world robotic platforms, we show three distinctive mechanisms for generating different gait patterns in these robots.

1 Introduction

Biological systems are capable of adaptive locomotion to traverse complex terrains efficiently and robustly by skillfully manipulating their musculoskeletal systems. In order to understand the underlying mechanisms of animals' agility, dexterity, and efficiency in motor control, there has been increasing interest in the study of gait patterns in biological and artificial legged systems. Previously, a number of different approaches have been proposed for this purpose, which can be roughly classified into the investigations of gait patterns induced by the central pattern generator models [3, 4, 20–22, 40, 43], the multi-gait locomotion models based on mechanical dynamics [2–5, 7, 11, 14, 18, 23, 37, 41, 47], stability [6, 8, 19, 27, 35, 36, 42], energetics [1, 12, 25, 28, 29], and locomotion velocity [9, 10, 15, 24, 26, 46]. The underlying research challenge that is common to most of these research areas is to develop the guiding principles to harness complex mechanical and neuronal dynamics, because, compared to biological systems that are able to control complex bodies in dynamic environments, our models and robots are still considerably simpler, and we do not fully understand how to scale up the degrees of complexity.

In response to this challenging problem, a number of researchers have been attempting to construct a theoretical framework to systematically investigate the roles of mechanical dynamics in adaptive legged locomotion, and in this context, the concept of morphological computation was created [16, 17, 30, 32, 33]. This concept states that the morphology and mechanical dynamics of autonomous systems play essential roles, and the specifics of mechanical designs have to be exploited in motor control of autonomous systems if they have to deal with uncertain dynamic environments. The previous case studies have detailed this concept in terms of self-stability of underactuated systems [18, 31, 34] and the real-world learning of complex motions such as biped walking [8, 44, 45], for example. While the concept of morphological computation has led to high-impact demonstrations, it turns out that one of the major problems in this line of research lies in the highly complex design processes of mechanical dynamics, and we do not have a systematic methodology to structure the system with a significant complexity to exhibit a large variety of behavioral patterns. As a result, most real-world locomotion robots are able to generate only one type of behavior (e.g., either walking or running), and it is still a significant challenge to increase the variations of gait patterns while exploiting mechanical dynamics.

From this perspective, we have been investigating an alternative approach to generate gait patterns while exploiting mechanical dynamics in order to develop a set of design principles. This approach makes use of free vibration of mechanical structures composed of curved elastic beams, and the beam is shaped into a specific structure such that free vibration induced by a rotating mass results in different locomotion patterns [39]. This approach demonstrated hopping locomotion by exploiting free vibration of a C-shaped elastic beam [38], and we also found that there are many advantages of using free vibration of the elastic beam for designing locomotion robots. First, an elastic beam has low damping losses, and its vibration with a large amplitude can be induced by a small power of actuation when operated at a resonance frequency. Second, the control of dynamic behavior in the elastic beam can be as simple as providing a basic oscillation of small masses. And third, design and assembly of this type of robot are very simple and have very low cost, which allows us to explore many different variations of structures to develop the design principles.

Based on these accumulated techniques and exploration so far, the main goal of this article is to propose a set of design principles that systematically explain robot locomotion based on free vibration, specifically for generating multiple gait patterns. We develop a conceptual model of locomotion robots, and explore a specific set of design parameters to explain how morphological designs and locomotion dynamics are related. Through experiments in simulation and real-world robots, we extract three different ways to control gait patterns of these robots, and based on the experimental results, we discuss the implications of morphological computation in the context of the proposed robot locomotion scheme.

We have structured this article as follows. First, in Section 2, we present the conceptual models of locomotion robots and controllers. Then, in Section 3, the experimental setup and simulation model are described. Section 4 presents the comparison of experimental and simulation results for four different gaits. And finally, in Section 5, we discuss the implications and challenges of this approach and make concluding remarks.

2 Conceptual Model of Multi-Gaited Robot

This article introduces a series of locomotion robots that were designed and constructed based on a single design concept, which is explained in this section. For systematic exploration and analysis, we describe the conceptual models of mechanical structures and motor control, and define some parameters to characterize the basic locomotion dynamics in this framework.

2.1 Model of Mechanical Structures

Each of the robots presented in this article consists of a DC motor, two large feet, and a main body made of a single-piece elastic beam as shown in Figure 1a. The U-shaped beam constitutes the leg and spine of the robot, and a rotating mass is attached to the motor, which is mounted at the center of the spine. The actuator controls the direction and magnitude of angular velocity of the rotating mass (for more details see Section 2.4). We used two large feet to ensure that the motion of the robot is in the sagittal plane, and also provided rubber material at the ground contact of the feet in order to gain sufficient ground friction.

Figure 1. 

(a) Prototype of the U-shaped elastic beam robot. (b) Schematic illustration and nomenclature of the robot. The curvature of the leg is expressed by c. (c) Conceptual model of the robot consists of three rigid links (legs and spine) and three point masses denoted by black circles. The spine and legs are connected via torsional springs—dampers of stiffness kθ and damping dθ. The legs of the robot are defined as linear (longitudinal) springs—dampers of stiffness kL and damping dL. The DC motor is fixed at the center of the spine. A bar of length R, with a point mass mR attached at the end, is actuated at an angular velocity ω. The masses of front foot and hind foot are represented by mf and mh. The robot has 17 state variables, namely, xf, yf, xh, yh (coordinates of front foot and hind foot), f, h, θh, θf (leg lengths and the angles between the legs and a line perpendicular to the spine), the corresponding velocities of those, and β (the angle between the rotating bar and spine).

Figure 1. 

(a) Prototype of the U-shaped elastic beam robot. (b) Schematic illustration and nomenclature of the robot. The curvature of the leg is expressed by c. (c) Conceptual model of the robot consists of three rigid links (legs and spine) and three point masses denoted by black circles. The spine and legs are connected via torsional springs—dampers of stiffness kθ and damping dθ. The legs of the robot are defined as linear (longitudinal) springs—dampers of stiffness kL and damping dL. The DC motor is fixed at the center of the spine. A bar of length R, with a point mass mR attached at the end, is actuated at an angular velocity ω. The masses of front foot and hind foot are represented by mf and mh. The robot has 17 state variables, namely, xf, yf, xh, yh (coordinates of front foot and hind foot), f, h, θh, θf (leg lengths and the angles between the legs and a line perpendicular to the spine), the corresponding velocities of those, and β (the angle between the rotating bar and spine).

To construct a conceptual model of these robots for a systematic analysis in simulation, we made the following assumptions:

Assumption 1. The behavior of this robot can be analyzed in the sagittal plane, and the ground contact of each of the feet can be treated as a point contact.

Assumption 2. The spine of the robot can be assumed to be a rigid beam, and the masses of spine and motor can be combined into a point mass at the center of the spine.

Assumption 3. The masses of the legs are negligible, and each of the feet can be represented by a point mass.

Assumption 4. The rotation between the leg and the spine can be represented by a torsional spring-damper element as shown in Figure 1c.

Assumption 5. The longitudinal deflection of the legs can be represented by a linear spring-damper element.

Assumption 6. The torsional and longitudinal stiffnesses of the above are assumed to be linear and do not vary, regardless of the shape and deflection of the robot body structures.

Assumption 7. The robot actuator can be modeled as a velocity-controlled motor (represented by the motor block of the SimMechanics simulator; see Section 2.4 for more details).

Based on these assumptions, our model consists of three rigid links connected to each other through torsional and linear spring-damper elements. There are three point masses on these links: First, the masses of actuator and spine are represented by a point mass mtop, which is located at the center of the spine. The other two point masses represent the masses of front foot and hind foot, which are denoted by mf and mh. The shape of the elastic beam can be defined by the natural lengths of the legs (i.e., the leg lengths when they are unloaded; f0 = h0 = 0); the curvature of the legs, c; the spine length s; and the rest angles between the leg and line perpendicular to the spine (θf0 = θh0 = θ0). When the elastic beam is without any compression or extension, the natural lengths of the legs and the rest angles of the symmetric robot are denoted by 0 and θ0. The torsional stiffness and damping of the two joints are represented by kθ and dθ, respectively. The leg deformation is represented by a linear spring with stiffness k, damping d, and natural length 0. The rotating mass mR is connected to the actuator by a bar of length R as shown in Figure 1c.

In summary, this model consists of 15 mechanical design parameters (k, d, kθ, dθ, θf0, θh0, f0, h0, s, c, mh, mf, mtop, R, and mR), one control parameter (ω; see details in Section 2.4), and 18 state variables (xf, yf, xh, yh, f, h, θf, θh, β, and corresponding velocities), as shown in Figure 1c and the  Appendix. All of the analysis of the conceptual model was conducted by using MATLAB (Mathworks, Inc.) together with the SimMechanics toolbox.

2.2 Analysis of Resonance Frequency

At a later stage in this article, we will analyze dynamic behaviors of the U-shaped elastic beam robot with respect to the resonance frequencies. We usually observe free vibration at two distinct oscillation frequencies: the torsional and longitudinal resonances. The first mode of vibration appears at a lower oscillation frequency ωθ, and it exhibits torsional oscillation trajectories. The second mode of vibration occurs at a higher oscillation frequency ωL, and it exhibits oscillation trajectories in the longitudinal direction.

The resonance frequencies are estimated both experimentally and analytically in this article. For the experimental estimation, we tested real-world robots and visually measured resonance frequencies when actuated (see Section 4.1 and Table 2, discussed later in Section 2.5). The analytical solutions can be derived from the conceptual model introduced in Section 2.1. Here, as the first estimate of resonance frequencies, we assume no damping in the springs, and the rest angles of both legs are equal to zero (θ0 = 0 (rad)).

2.2.1 Resonance Frequency for the Torsional Motion

One of the simplest analytical solutions for the resonance frequency in the proposed system can be computed when we assume the entire robot as a bridgelike rigid structure with a pair of upright supporting legs in parallel (θh = θf = θ, f = h = ). In this case, the Lagrangian can be derived as
formula
By differentiating the Lagrangian with respect to the torsion angle θ, the torsional equation of the motion can be written as
formula
formula
For small angles (sinθ ≈ θ), Equation 3 can be linearized and the torsional resonance frequency ωθ of the U-shaped elastic beam can be obtained by using the homogeneous part of this differential equation:
formula
formula
where kθ represents the torsional stiffness of the beam at the intersegmental joints between the spine and the legs, as shown in Figure 1c.

2.2.2 Resonance Frequency for the Longitudinal Motion

When the Lagrangian in Equation 1 is differentiated with respect to the leg length , we obtain
formula
formula
The longitudinal resonance frequency (ωL) of the elastic beam can be obtained by using the homogeneous part of this equation:
formula
formula
where kL represents the longitudinal stiffness of the legs as shown in Figure 1c.

2.3 Parameterizing the Morphology

According to the conceptual model in Section 2.1, the shape of the elastic beam can be characterized by three parameters, namely the natural length of the robot legs, 0; the length of the spine, s; and the curvature of the leg, c, as shown in Figure 1b. From Equation 5, we can also see that the resonance frequency of the robot is essentially determined by the leg length .

To facilitate the analysis later, here we define two geometric design parameters: the shape ratio rS and curvature ratio rc. The shape ratio is defined as
formula
The curvature ratio is the ratio of the radius of curvature of the leg, c, to the natural leg length of the robot, 0:
formula

The effect of these geometric design parameters on gait patterns of the robot is discussed later, in Section 4.

2.4 Motor Control Model

In order to systematically analyze the intrinsic body dynamics derived from the morphological properties, we employed a minimalistic control strategy: Each of the robots presented in this article uses only one DC motor without any sensory feedback, and the motor is activated by a constant voltage during every locomotion experiment. Moreover, in each robot platform, the DC motor contains a large gear reduction (see also Section 3.1 for more details); thus the angular velocity of the rotating mass can be assumed to be largely unchanged when a constant voltage is applied to the motor.

To construct a model that characterizes the actuation in the real-world robots, we consider the centripetal force generated by the rotating mass: Assuming the angular velocity of the rotating mass to be constant, the centripetal force F(t) can be expressed in terms of the angular velocity ω, the rotating mass mR, and the radius of rotation R as follows:
formula
From this equation, we see that locomotion patterns of our robots can be influenced not only by the angular velocity, but also by the other design parameters, namely, the rotating mass and the rotation radius.

2.5 Ground Interaction Model

For the simulation experiments shown later in this article, we employ a physically realistic interaction model based on [13]. The vertical ground interaction forces are approximated by nonlinear spring-damper interactions, and the horizontal forces are calculated by a sliding-stiction model. The model switches between sliding and stiction when the velocity of the foot becomes lower or higher than the specified limit determined by the sliding and stiction friction coefficients, μslide and μstick, respectively. The vertical (Gyi) and horizontal (Gxi) ground reaction forces are computed as
formula
formula
where and yci denote the horizontal velocity and the vertical distance of the contact point i from the ground surface, respectively, and Fxci denotes the computed force at the foot contact point i. We used the following parameters to simulate the ground interaction: a = 2.5 × 105 N/m3, b = 3.3 s/m. The friction coefficients (μslide, μstick) and the damping coefficients (dθ, dL) of the springs are selected such that the simulation model can represent the dynamics of different gait patterns observed in the real world. The specifications for this simulation model are given in Tables 1 and 2.
Table 1. 

Specifications for real-world robots.

0 0.43 m R 0.15 m rc 0.15 θ0 0.07 (rad) 
mh 0.12 kg mf 0.07 kg mR 0.025 kg mtop 0.12 kg 
kL 200 N/m dL 0.01 N s/m kθ 4 N m/rad dθ 0.01 N m s/rad 
μstick 0.6 μslide 0.5 
0 0.43 m R 0.15 m rc 0.15 θ0 0.07 (rad) 
mh 0.12 kg mf 0.07 kg mR 0.025 kg mtop 0.12 kg 
kL 200 N/m dL 0.01 N s/m kθ 4 N m/rad dθ 0.01 N m s/rad 
μstick 0.6 μslide 0.5 
Table 2. 

Variations in real-world robots.

Robot ID number
Leg length 0 (m)
Shape ratio rS = s/0
ωθ (rad/s)
ωL (rad/s)
0.43 0.35 16.6 38 
0.43 0.45 14.7 37 
0.43 0.55 14.2 36 
0.43 0.65 14.3 36 
0.43 0.80 14.1 36 
0.43 0.90 14.1 35 
0.43 1.00 14.0 35 
0.43 1.10 14.0 34 
0.43 1.20 13.9 34 
10 0.43 1.30 13.9 34 
Robot ID number
Leg length 0 (m)
Shape ratio rS = s/0
ωθ (rad/s)
ωL (rad/s)
0.43 0.35 16.6 38 
0.43 0.45 14.7 37 
0.43 0.55 14.2 36 
0.43 0.65 14.3 36 
0.43 0.80 14.1 36 
0.43 0.90 14.1 35 
0.43 1.00 14.0 35 
0.43 1.10 14.0 34 
0.43 1.20 13.9 34 
10 0.43 1.30 13.9 34 

3 Experiments

This section describes the experiments using real-world robots. First, we introduce the method of experimentation, and then, we explain the basic observations of the locomotion behaviors.

3.1 Experimental Setup

The main goal of the experiments is to systematically explore the influences of morphology and motor control on locomotion behaviors in real-world platforms. For this purpose, we developed 10 platforms with different shape ratios rS, and each of the platforms is labeled by a robot ID number as shown in Figure 2. Except for the shape ratio, all the other specifications are identical.

Figure 2. 

(a) Photograph of the variations of the U-shaped elastic beam along with robot numbers. (b) Real-world robots with different shape ratios (rS ∈ [0.35, 1.3]).

Figure 2. 

(a) Photograph of the variations of the U-shaped elastic beam along with robot numbers. (b) Real-world robots with different shape ratios (rS ∈ [0.35, 1.3]).

Every robot contains a DC motor with a large gear reduction (50:1 micro metal gearmotor produced by Pololu Robotics and Electronics). During the experiments, the motor is powered externally with a constant voltage. In every experiment, we conducted multiple trials with different voltages in order to identify the angular velocity of the rotating mass for stable robot locomotion.

All of the locomotion experiments shown in this article were conducted on a 4-m-long flat wooden floor. The experimental arena was recorded by both a standard video camera at 30 frames per second and a high-speed camera at 120 frames per second in order to analyze dynamic motions of the robots. The recorded data was postprocessed after the experiments to extract the experimental results presented in Section 4.

3.2 Gait Patterns

On applying the measured constant voltage to the motor, each of our robot platforms exhibits variations of gait patterns on a flat ground. In this section, we explain the basic characteristics of the four stable gait patterns that we found in the real-world experiments1 (see also Figures 3 and 4).

Figure 3. 

Different locomotion gaits observed in the U-shaped elastic beam robot. These four gaits differ from each other according to the number of phases of locomotion. The simulation results are shown by stick diagrams, where the dashed lines represent that the ground reaction force of the leg is zero. Time-series photographs of real-world robot for one cycle of rotation during (a) gait 1 (robot ID 9), (b) gait 2 (ID 4), (c) gait 3 (ID 4), (d) gait 4 (ID 4).

Figure 3. 

Different locomotion gaits observed in the U-shaped elastic beam robot. These four gaits differ from each other according to the number of phases of locomotion. The simulation results are shown by stick diagrams, where the dashed lines represent that the ground reaction force of the leg is zero. Time-series photographs of real-world robot for one cycle of rotation during (a) gait 1 (robot ID 9), (b) gait 2 (ID 4), (c) gait 3 (ID 4), (d) gait 4 (ID 4).

Figure 4. 

Simulation results shown in gait diagrams of the multi-gaited U-shaped elastic beam robot for two cycles of rotation of the rotating mass. The shaded region indicates that a foot is touching the ground. (a) Gait 1, 0–100% double-stance phase; (b) gait 2, 0–30% double-stance phase, 35–95% flight phase; (c) gait 3, 49–80% front leg stance, 41–49% and 80–88% double-stance phase, 88–141% hind leg stance; (d) gait 4, 18–31% flight phase, 31–72% hind leg stance, 72–86% double-stance phase, 86–118% front leg stance.

Figure 4. 

Simulation results shown in gait diagrams of the multi-gaited U-shaped elastic beam robot for two cycles of rotation of the rotating mass. The shaded region indicates that a foot is touching the ground. (a) Gait 1, 0–100% double-stance phase; (b) gait 2, 0–30% double-stance phase, 35–95% flight phase; (c) gait 3, 49–80% front leg stance, 41–49% and 80–88% double-stance phase, 88–141% hind leg stance; (d) gait 4, 18–31% flight phase, 31–72% hind leg stance, 72–86% double-stance phase, 86–118% front leg stance.

3.2.1 Gait 1

This gait pattern can typically be observed in a platform with a long spine link when it is actuated at a low angular velocity of the rotating mass (see Section 4 for more details). As shown in Figure 3a, the locomotion of gait 1 is based on the torsional vibration of the robot and the pair of legs oscillate in phase in the fore-aft plane. The direction of rotation of the mass determines the direction of locomotion; counterclockwise rotation resulted in locomotion toward the right-hand side in Figure 3a. The robot platform usually exhibits sliding ground interaction throughout the entire cycle of gait 1; thus we observe only one phase in the gait diagram (Figure 4a).

3.2.2 Gait 2

This gait pattern is usually observed in a platform with a shorter spine link, and the angular velocity of the rotating mass needs to be very large compared to those of all other gaits. Gait 2 is a hopping type of locomotion, in which the whole body of the robot oscillates vertically with a large amplitude. As a result, gait 2 shows clear flight phases in the gait diagram (Figure 4b) with both legs taking off and landing at the same time. The direction of locomotion is also determined by the direction of rotation as in gait 1, and the clockwise rotation in Figure 3b resulted in locomotion toward the right-hand side.

3.3.3 Gait 3

Gait 3 is also observed in a platform that has a shorter spine and whose rotating mass is typically actuated at a lower angular velocity. This locomotion dynamics shows a walking-like gait in which we find rotary transitions between three phases (front-leg stance, hind-leg stance, and double stance) without a flight phase, as shown in Figure 4c. The direction of rotation of the mass has to be clockwise as shown in Figure 3c, and reversal of the direction results in unstable locomotion or a transition to gait 4.

3.3.4 Gait 4

This gait is usually observed in a platform that has a spine length in the middle range (see Section 4 for more details). The rotating mass is also actuated at a lower velocity, and its direction has to be counterclockwise, as shown in Figure 3d. This gait pattern is similar to a bounding gait in quadruped animals, and it shows four phases (front stance, hind stance, double stance, and flight).

4 Mechanisms of Gait Generation

In this section, we will explore three underlying mechanisms to generate the different gait patterns: through variations of resonance frequencies, shape ratios, and motor control. By analyzing the experimental results in the real world and in simulation, we will explain the influences of those mechanisms on the gait patterns of our robots.

4.1 Effect of Resonance Frequency

The resonance frequencies of the robot body structures characterize the basic locomotion dynamics in the proposed approach. As explained in Sections 2 and 3, here we first consider the influences of the torsional and longitudinal resonance frequencies on locomotion behaviors, and for this purpose, we conducted two sets of experiments.

The first series of experiments were conducted in order to measure the resonance frequencies of the robot platforms. In these experiments, we measured the torsional resonance frequency ωθ by applying a horizontal force to the robot structure. The force applied is such that the feet of the robot do not slide on the ground. We recorded this experiment using a high-speed camera at 120 frames per second and measured the torsional oscillation with respect to time to calculate ωθ. For the longitudinal resonance frequency ωL, we applied a vertical force on the spine of the robot and measured the longitudinal deflection of the spine. Later, we calculated the longitudinal stiffness kL of the robot, which is the ratio of applied force to deflection. By using Equation 9, the longitudinal resonance frequency is calculated for each robot.

The second series of experiments were conducted by using all 10 robots, and we analyzed the experimental results in terms of shape ratio and resultant gait patterns. More specifically, the experiments were conducted on each of 10 robots by varying the constant voltage applied to the motor. On each robot, we obtained the voltage values at which the best locomotion performance could be observed. Typically we observed multiple gait patterns in most of the robots; therefore, we identified the voltage values that induce the most stable gait patterns.

In Figure 5, we have plotted the experimental results of resonance frequencies of the 10 robots as well as the angular velocities of their locomotion. This figure shows three important characteristics of the proposed approach. First, while all four gait patterns are shown, none of these robots was able to show all gait patterns, and only two robots out of the 10 could exhibit three gait patterns, namely, the robots with shape ratio 0.55 and 0.65. Other than those, three robots showed two gait patterns, four robots showed only one, and one of the robots did not show any stable pattern. Second, the locomotion behaviors of all of these robots are achieved at angular velocities that are close to, or slightly lower than, the resonance frequencies. Although we do not know the underlying relationship between the locomotion frequency and resonance frequency in detail, it seems that the resonance frequencies could be a good indicator of angular velocity in the proposed approach. And third, there are two clearly distinguished groups of gait patterns: Namely, gait 2 is clearly separated from the other three gaits, which require a larger angular velocity that is closer to the longitudinal resonance frequency ωL.

Figure 5. 

Experimental results on angular velocity ω for achieving different gait patterns for 10 different robots. Gait 2 is achieved when the angular velocity is close to the longitudinal resonance frequency ωL of the robot. Gait 1, gait 3, and gait 4 are achieved when the angular velocity is close to the torsional resonance frequency ωθ.

Figure 5. 

Experimental results on angular velocity ω for achieving different gait patterns for 10 different robots. Gait 2 is achieved when the angular velocity is close to the longitudinal resonance frequency ωL of the robot. Gait 1, gait 3, and gait 4 are achieved when the angular velocity is close to the torsional resonance frequency ωθ.

From these experiments, in order to specify the angular velocity of gait 2, we found that the curvature ratio rc in Equation 11 plays an important role. Here we constructed five different beams with different curvature ratios (rc = 0.15, 0.25, 0.35, 0.5, 0.7), and experimentally investigated the effect of this parameter with respect to the torsional and longitudinal resonance frequencies ωθ and ωL. As shown in Figure 6, a larger curvature ratio rc provides a lower resonance frequency for both torsional and longitudinal directions, although the latter effect is much more significant. Therefore, if we design the robot with a low curvature ratio rc—say, rc = 0.15—we can predict that the longitudinal resonance frequency should be 60 rad/s while the torsional one should be approximately 18 rad/s.

Figure 6. 

The effect of the curvature ratio rc of the leg on the resonance frequency. (a) Torsional resonance frequency ωθ is a linear function of curvature ratio. (b) Longitudinal resonance frequency ωL is a quadratic function of curvature ratio.

Figure 6. 

The effect of the curvature ratio rc of the leg on the resonance frequency. (a) Torsional resonance frequency ωθ is a linear function of curvature ratio. (b) Longitudinal resonance frequency ωL is a quadratic function of curvature ratio.

In conclusion, from the analysis in this section, we learned that the shape ratio and curvature ratio are critical parameters that determine the operation frequency of locomotion as well as the number of gait patterns that a robot can generate. With a proper choice of these two parameters, a robot can be made to exhibit as many as three gait patterns just by varying the velocity of the rotating mass. Another important implication lies in the fact that the shape ratio can be another important parameter in increasing the variety of gait patterns: If a robot were able to vary its shape ratio on the fly (our current robots cannot achieve this), it could achieve as many as four gait patterns, which is not possible otherwise.

4.2 Effect of Shape Ratio rS

This section explores the influence of the shape ratio rS on the gait patterns and locomotion velocity of our robots. To analyze the velocity, we employ a dimensionless measure, the so-called Froude number:
formula
By using this measure, we continued the analysis of our locomotion experiments in the previous section, and plotted the results with respect to Froude number and the shape ratios as shown in Figure 7a. In addition, to confirm the experimental results, we also conducted the same tests by using the simulation model explained in Section 2 and plotted the results in Figure 7b.
Figure 7. 

Variation of Froude number with the shape ratio: (a) experimental results, (b) simulation results. The shaded areas in both graphs represent the range of shape ratio in which we found no stable locomotion.

Figure 7. 

Variation of Froude number with the shape ratio: (a) experimental results, (b) simulation results. The shaded areas in both graphs represent the range of shape ratio in which we found no stable locomotion.

One of the important characteristics that is common to the real-world and simulation experiments is that the maximum Froude number is achieved in different gait patterns, depending on the shape ratio. More specifically, with a small shape ratio rS between 0.35 and 0.65, gait 3 exhibits the largest Froude numbers, but that maximum shifts to gait 4 and gait 1 as the shape ratio increases. This result implies that fast locomotion does not necessarily require running-like locomotion with flight phases (gait 4), and it largely depends on the shape ratio. Having said that, another important finding in these figures is that the maximum Froude numbers in both real-world and simulated robots were achieved in gait 4 when the robot was given the optimum shape ratio (viz., rc between 0.8 and 0.9).

Considering the experimental results, we realize that the maximum Froude numbers are always achieved at an angular velocity of the rotating mass near to the torsional resonance frequency, regardless of the shape ratio, and it is not necessary to use fast and high-power actuation of the rotating mass to induce rapid locomotion. More specifically, in the lower range of shape ratios, gait 3 was achieved at an angular velocity of about 10 rad/s (in Figure 5); in the middle range, gait 4 was achieved at 15 rad/s; and in the higher range, gait 1 was operated at about 12 rad/s.

4.3 Effect of Motor Control

So far we have identified how two gait patterns (viz., gaits 1 and 2) can be independently induced through the angular velocity of the rotating mass and the shape ratio: With a larger shape ratio (rS > 1.2), the robot exhibits gait 1 exclusively; and with a smaller shape ratio and a higher angular velocity, the robot exhibits only gait 2.

In order to generate the different dynamics for gaits 3 and 4, both of which are induced by similar ranges of angular velocity of rotating mass and shape ratios, we need to introduce another motor control parameter, namely, the direction of rotating-mass actuation. As shown in Figure 3c and 3d, the rotating mass is actuated clockwise for gait 3, and counterclockwise for gait 4.

To analyze the influence of the direction of rotation, here we define another dimensionless quantity Fmv, which is the ratio of the vertical component of centripetal force to the total weight of the robot:
formula
where β denotes the angle of the rotating mass with respect to the spine of the robot (see Figure 1c). We compared the variation of this dimensionless number during the stable locomotion in gaits 3 and 4 in simulation, and the results are plotted in Figure 8. Figure 8a and 8c show the order and timing of different phases with respect to the angle of rotating mass during one cycle, and Figure 8b and 8d illustrate the variation of the dimensionless force Fmv. These figures were based on the simulation experiments, but we confirmed similar behaviors in the real-world robots.
Figure 8. 

Simulation results on motor control and gait patterns. (a) The angle of the rotating mass, β, with three different phases in gait 3. (b) The dimensionless number Fmv for one cycle of gait 3. The shaded areas correspond to the locomotion phases in (a). (c) Gait 4 with four different phases of locomotion. (d) The Fmv for one cycle of gait 4, and the shaded areas correspond to the locomotion phases in (c).

Figure 8. 

Simulation results on motor control and gait patterns. (a) The angle of the rotating mass, β, with three different phases in gait 3. (b) The dimensionless number Fmv for one cycle of gait 3. The shaded areas correspond to the locomotion phases in (a). (c) Gait 4 with four different phases of locomotion. (d) The Fmv for one cycle of gait 4, and the shaded areas correspond to the locomotion phases in (c).

From these experimental results, we notice two factors that differentiate the basic dynamics of gaits 3 and 4. First, by comparing Figure 8a and 8c, we observe that there is a long hind-leg stance phase in gait 3 (Figure 8a), which covers more than half of the entire gait cycle, whereas the proportion of front- and hind-leg stance phases in gait 4 is almost the same. For example, when the rotating mass is at 75% in gait 3, the robot is in the hind-leg stance phase, although it is in a double-stance phase when the rotating mass is at 25%. In contrast, the periods of different phases in gait 4 are clearly more symmetric. Thus we see that gait 4 relies on the flight phase to achieve locomotion. This contrast essentially implies that the underlying mechanical dynamics of gaits 3 and 4 are different, although they appear to be similar in the gait diagrams in Figure 4c and 4d. And second, it is also important to note that, if we compare Figure 8b and 8d, the amplitude of Fmv is significantly larger in gait 3 than in gait 4, which can be explained by the larger angular velocity of the rotating mass in gait 4 (as shown in Figure 5). The Fmv value essentially indicates the vertical component of forces induced by the rotating mass; thus a large positive value of Fmv induces a large acceleration, leading to a flight phase of the whole robot.

5 Discussion and Conclusion

This article has presented a novel approach to exploit the intrinsic mechanical dynamics of elastic beams for the purpose of robot locomotion. In our approach, the robot is composed of an elastic beam and a small rotating mass, which triggers a forced oscillation of the entire body structure at one of the resonance frequencies. By designing the body structure such that the free vibration could result in a basic locomotion dynamics, the robot is enabled not only to achieve stable and efficient locomotion, but also to exhibit different gait patterns such as walking, hopping, or running. Furthermore, because the robot is composed of elastic beams, we are able to systematically analyze the influence of shapes and the other design parameters on the overall locomotion behaviors. We introduced four distinctive gait patterns that the robots are able to generate, and through simulation and real-world experiments, we identified three underlying mechanisms (the resonance frequency, the shape ratio, and control of the rotating mass) with which the different gait patterns can be controlled.

Unlike the conventional methods of designing and constructing locomotion robots, our approach allows us to systematically develop and analyze relatively complex mechanical dynamics depending on the free vibration of the robot's entire body without heuristically searching the space of design and control parameters. For example, in the analysis of our robot behaviors, we did not consider the body of the elastic beam as a continuum, but approximated it by a combination of simple linear and rotational springs. Even with the simplified model, the simulation and experimental results on overall behaviors matched reasonably well. The analysis method introduced in this article is particularly important for increasing the level of complexity in dynamic robot locomotion, because it implies that the overall dynamics of a complex body structures may be reducible to a simplified model for ease of analysis and control. As a matter of fact, this article has demonstrated how complex gait patterns can be controlled by attention to their underlying mechanisms.

By extending the proposed approach, we expect to identify more advanced methodology to systematically analyze and develop complex dynamics for the purpose of morphological computation research. In addition, we are planning to investigate further the effect of multi-gaited locomotion on energy efficiency. Because the proposed approach scales across different sizes, weights, and velocities, we expect that the robot is able to exhibit gait patterns even under a large variety of physical conditions, which should be a highly interesting research topic in connection with the abilities of biological systems. Finally, it is also important to develop more precise physical models of the robot platforms and dynamics of gait patterns that were not explored in this article. Such modeling will provide additional insights into the underlying mechanisms of gait patterns as well as more advanced control architectures.

Acknowledgments

This work was supported by the Swiss National Science Foundation, Grant No. PP00P2123387. This research is also funded by the Swiss National Science Foundation through the National Centre of Competence in Research Robotics, and by the Technological Research Council of Turkey, TUBITAK.

Notes

1 

See the gait demonstration videos of real-world and simulated robots at http://www.birl.ethz.ch/Robots (Multi-Gaited Robot).

References

1. 
Ahmadi
,
M.
, &
Buehler
,
M.
(
2006
).
Controlled passive dynamic running experiments with the ARL-monopod II.
IEEE Transactions on Robotics
,
22
(
5
),
974
986
.
2. 
Akinfiev
,
T.
,
Armada
,
M.
, &
Montes
,
H.
(
2003
).
Vertical movement of resonance hopping robot with electric drive and simple control system.
In
Proceedings of the 2003 IEEE Mediterranean Conference on Control and Automation
.
3. 
Aoi
,
S.
,
Yamashita
,
T.
,
Ichikawa
,
A.
, &
Tsuchiya
,
K.
(
2010
).
Hysteresis in gait transition induced by changing waist joint stiffness of a quadruped robot driven by nonlinear oscillators with phase resetting.
In
Proceedings of the 2010 IEEE International Conference on Intelligent Robots and Systems
(pp.
1915
1920
).
4. 
Aoi
,
S.
,
Yamashita
,
T.
, &
Tsuchiya
,
K.
(
2011
).
Hysteresis in the gait transition of a quadruped investigated using simple body mechanical and oscillator network models.
Physical Review E, Statistical, Nonlinear, and Soft Matter Physics
,
83
(
6, Pt 1
),
061909
.
5. 
Blickhan
,
R.
(
1989
).
The spring-mass model for running and hopping.
Journal of Biomechanics
,
22
,
1217
1227
.
6. 
Blickhan
,
R.
,
Seyfarth
,
A.
,
Geyer
,
H.
,
Grimmer
,
S.
,
Wagner
,
H.
, &
Gunther
,
M.
(
2007
).
Intelligence by mechanics.
Philosophical Transactions of the Royal Society, A
,
365
,
199
220
.
7. 
Chatterjee
,
A.
,
Pratap
,
R.
,
Reddy
,
C. K.
, &
Ruina
,
A.
(
2002
).
Persistent passive hopping and juggling is possible even with plastic collisions.
The International Journal of Robotics Research
,
21
,
621
634
.
8. 
Collins
,
S.
,
Ruina
,
A.
,
Tedrake
,
R.
, &
Wisse
,
M.
(
2005
).
Efficient bipedal robots based on passive dynamic walkers.
Science
,
307
,
1082
1085
.
9. 
Coros
,
S.
,
Karpathy
,
A.
,
Jones
,
B.
,
Reveret
,
L.
, &
Panne
,
M.
(
2011
).
Locomotion skills for simulated quadrupeds.
ACM Transactions on Graphics
,
30
(
4
),
59
.
10. 
Diedrich
,
F. J.
, &
Warren
,
W. H.
(
1990
).
Trotting, pacing and bounding by a quadruped robot.
Journal of Biomechanics
,
23
,
79
98
.
11. 
Diedrich
,
F. J.
, &
Warren
,
W. H.
(
1995
).
Why change gaits? Dynamics of the walk-run transition.
Journal of Experimental Psychology
,
21
(
1
),
183
202
.
12. 
Gabrielli
,
G.
, &
von Karman
,
T.
(
1950
).
What price speed?
Mechanical Engineering
,
72
,
775
781
.
13. 
Gerritsen
,
K. G. M.
,
van den Bogert
,
A. J.
, &
Nigg
,
B. M.
(
1995
).
Direct dynamics simulation of the impact phase in heel-toe running.
Journal of Biomechanics
,
28
(
6
),
661
668
.
14. 
Geyer
,
H.
,
Seyfarth
,
A.
, &
Blickhan
,
R.
(
2005
).
Spring-mass running: Simple approximate solution and application to gait stability.
Journal of Theoretical Biology
,
232
(
3
),
315
328
.
15. 
Han
,
B.
,
Luo
,
X.
,
Wang
,
X.
, &
Chen
,
X.
(
2011
).
Mechanism design and gait experiment of an amphibian robotic turtle.
Advanced Robotics
,
25
,
2083
2097
.
16. 
Hauser
,
H.
,
Ijspeert
,
A. J.
,
Füchslin
,
R. M.
,
Pfeifer
,
R.
, &
Maass
,
W.
(
2011
).
Towards a theoretical foundation for morphological computation with compliant bodies.
Biological Cybernetics
,
105
,
355
370
.
17. 
Iida
,
F.
, &
Pfeifer
,
R.
(
2006
).
Sensing through body dynamics.
Robotics and Autonomous Systems
,
54
(
8
),
631
640
.
18. 
Iida
,
F.
,
Rummel
,
J.
, &
Seyfarth
,
A.
(
2008
).
Bipedal walking and running with spring-like biarticular muscles.
Journal of Biomechanics
,
41
,
656
667
.
19. 
Iida
,
F.
, &
Tedrake
,
R.
(
2007
).
Motor control optimization of compliant one-legged locomotion in rough terrain.
In
Proceedings of the 2007 IEEE International Conference on Intelligent Robots and Systems
(pp.
2230
2235
).
20. 
Ijspeert
,
A. J.
(
2001
).
A connectionist central pattern generator for the aquatic and terrestrial gaits of a simulated salamander.
Biological Cybernetics
,
84
,
331
348
.
21. 
Ijspeert
,
A. J.
(
2008
).
Central pattern generators for locomotion control in animals and robots: A review.
Neural Networks
,
21
(
4
),
642
653
.
22. 
Ijspeert
,
A. J.
,
Crespi
,
A.
,
Ryczko
,
D.
, &
Cabelguen
,
J. M.
(
2007
).
From swimming to walking with a salamander robot driven by a spinal cord model.
Science
,
315
(
5817
),
1416
.
23. 
Kar
,
D. C.
,
Issac
,
K. K.
, &
Jayarajan
,
K.
(
2003
).
Gaits and energetics in terrestrial legged locomotion.
Mechanism and Machine Theory
,
38
,
355
366
.
24. 
Kokkevis
,
E.
,
Metaxas
,
D.
, &
Badler
,
N. I.
(
1995
).
Autonomous animation and control of four-legged animals
Graphics Interface
,
1995
,
10
.
25. 
Kuo
,
A. D.
(
2007
).
Choosing your steps carefully: Trade-offs between economy and versatility in dynamic walking bipedal robots.
IEEE Robotics, Automation Magazine
,
7
,
18
29
.
26. 
Leeser
,
K. F.
(
1996
).
Locomotion experiments on a planar quadruped robot with articulated spine
,
M.S thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA
.
27. 
McGeer
,
T.
(
1990
).
Passive dynamic walking.
The International Journal of Robotics Research
,
9
(
2
),
62
82
.
28. 
Owaki
,
D.
,
Koyama
,
M.
,
Yamaguchi
,
S.
,
Kubo
,
S.
, &
Ishiguro
,
A.
(
2010
).
A two-dimensional passive dynamic running biped with knees.
In
Proceedings of the 2010 IEEE International Conference on Robotics and Automation
(pp.
5237
5242
).
29. 
Owaki
,
D.
,
Koyama
,
M.
,
Yamaguchi
,
S.
,
Kubo
,
S.
, &
Ishiguro
,
A.
(
2011
).
A 2-D passive-dynamic-running biped with elastic elements.
IEEE Transactions on Robotics
,
27
(
1
),
156
162
.
30. 
Pfeifer
,
R.
,
Iida
,
F.
, &
Gomez
,
G.
(
2006
).
Morphological computation for adaptive behavior and cognition.
International Congress Series
,
1291
,
22
29
.
31. 
Pfeifer
,
R.
,
Iida
,
F.
, &
Bongard
,
J.
(
2005
).
New robotics: Design principles for intelligent systems.
Artificial Life
,
11
(
1–2
),
99
120
.
32. 
Pfeifer
,
R.
, &
Scheier
,
C.
(
1999
).
Understanding intelligence.
Cambridge, MA
:
MIT Press
.
33. 
Pfeifer
,
R.
, &
Bongard
,
J. C.
(
2006
).
How the body shapes the way we think.
Cambridge, MA
:
MIT Press
.
34. 
Pfeifer
,
R.
,
Lungarella
,
M.
, &
Iida
,
F.
(
2007
).
Self-organization, embodiment, and biologically inspired robotics.
Science
,
318
,
1088
1093
.
35. 
Raibert
,
M.
(
1986
).
Legged robots that balance.
Cambridge, MA
:
MIT Press
.
36. 
Raibert
,
M.
,
Brown
,
H.
, &
Chepponis
,
M.
(
1984
).
Experiments in balance with a 3D one-legged hopping machine.
The International Journal of Robotics Research
,
3
(
2
),
75
92
.
37. 
Raynor
,
A. J.
,
Yi
,
C. J.
,
Abernethy
,
B.
, &
Jong
,
Q. J.
(
2002
).
Are transitions in human gait determined by mechanical, kinetic or energetic factors?
Human Movement Science
,
21
,
785
805
.
38. 
Reis
,
M.
, &
Iida
,
F.
(
2011
).
Hopping robot based on free vibration of an elastic curved beam.
In
Proceedings of the 2011 IEEE/ASME International Conference on Advanced Intelligent Mechatronics
(pp.
892
897
).
39. 
Reis
,
M.
,
Maheshwari
,
N.
, &
Iida
,
F.
(
2011
).
Self-organization of robot walking and hopping based on free vibration.
In
Proceedings of the 2nd International Conference on Morphological Computation
(pp.
91
93
).
40. 
Righetti
,
L.
, &
Ijspeert
,
A. J.
(
2008
).
Pattern generators with sensory feedback for the control of quadruped locomotion.
In
Proceedings of the 2008 IEEE International Conference on Robotics and Automation
(pp.
819
824
).
41. 
Roberts
,
T. J.
,
Kram
,
R.
,
Weyand
,
P. G.
, &
Taylor
,
C. R.
(
1998
).
Energetics of bipedal running: I. Metabolic cost of generating force.
The Journal of Experimental Biology
,
201
,
2745
2751
.
42. 
Spong
,
M. W.
, &
Bhatia
,
G.
(
2003
).
Further results on control of the compass gait biped.
In
Proceedings of the 2003 IEEE International Conference on Intelligent Robots and Systems
(pp.
1933
1938
).
43. 
Taga
,
G.
,
Yamaguchi
,
Y.
, &
Shimizu
,
H.
(
1991
).
Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment.
Biological Cybernetics
,
65
(
3
),
147
159
.
44. 
Tedrake
,
R.
, &
Seung
,
H. S.
(
2002
).
Improved dynamic stability using reinforcement learning.
In
Proceedings of the 5th International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines (CLAWAR)
(pp.
341
348
).
45. 
Tedrake
,
R.
,
Zhang
,
T. W.
, &
Seung
,
H. S.
(
2004
).
Stochastic policy gradient reinforcement learning on a simple 3D biped.
In
Proceedings of the 2004 IEEE International Conference on Intelligent Robots and Systems
(pp.
2849
2854
).
46. 
Tsujita
,
K.
,
Tsuchiya
,
K.
, &
Onat
,
A.
(
2001
).
Adaptive gait pattern control of a quadruped locomotion robot.
The International Journal of Robotics Research
,
4
,
2318
2325
.
47. 
Zeglin
,
G.
(
1999
).
The bow leg hopping robot (CMU-RI-TR-99-33).
Ph.D. thesis, Carnegie Mellon University
.

Appendix

     
  • t

    Time (s)

  •  
  • xf, xh

    Horizontal positions of the front and the hind feet (m)

  •  
  • yf, yh

    Vertical positions of the front and the hind feet (m)

  •  
  • mtop

    Mass of the motor and spine (kg)

  •  
  • mf, mh

    Masses of the front and the hind feet (kg)

  •  
  • mR

    Rotating mass (kg)

  •  
  • M

    Total mass of the robot (kg)

  •  
  • Lf, Lh

    Length of the front and hind legs (m)

  •  
  • Ls

    Length of the spine (m)

  •  
  • Lc

    Horizontal length of the elliptic curvature of the legs (m)

  •  
  • L0

    Length of the legs with no load (m)

  •  
  • rc

    Curvature ratio Lc/L0

  •  
  • rs

    Shape ratio Lh/L0

  •  
  • R

    Radius of rotation of the rotating mass (m)

  •  
  • θf, θh

    Offset angles of the front and hind legs (rad)

  •  
  • θ0

    Offset angle of the legs with no load (rad)

  •  
  • β

    Angle of the rotating mass with respect to ground (rad)

  •  
  • kL

    Longitudinal stiffness of the curved beam (N/m)

  •  
  • dL

    Damping coefficient of the longitudinal stiffness

  •  
  • kθ

    Torsional stiffness of the beam (N m/rad)

  •  
  • dθ

    Damping coefficient of the torsional stiffness

  •  
  • ω

    Angular velocity of the rotating mass (rad/s)

  •  
  • ωL

    Longitudinal resonance frequency of the U-shaped beam (rad/s)

  •  
  • ωθ

    Torsional resonance frequency of the U-shaped beam (rad/s)

  •  
  • F

    Centripetal force on the rotating mass (N)

  •  
  • Fmv

    Vertical component of normalized centripetal force

  •  
  • V

    Gait speed of the robot (m/s)

  •  
  • E

    Modulus of elasticity (N/m2)

  •  
  • I

    Moment of inertia (kg m2)

  •  
  • g

    Acceleration of gravity (m/s2)

  •  
  • Gxi

    Horizontal component of the ground reaction force (N)

  •  
  • Gyi

    Vertical component of the ground reaction force (N)

  •  
  • Horizontal velocity of the foot that makes contact with the ground (m/s)

  •  
  • Vertical velocity of the foot that makes contact with the ground (m/s)

  •  
  • yci

    Vertical distance of the contact point from ground surface (m)

  •  
  • μslide

    Sliding friction coefficient

  •  
  • μstick

    Stiction friction coefficient

  •  
  • Lagrangian (J)

Author notes

Contact author.

∗∗

Bio-Inspired Robotics Lab, Institute of Robotics and Intelligent Systems, Swiss Federal Institute of Technology Zurich, LEO D, Leonhardstrasse 27, 8092 Zurich, Switzerland. E-mail: murat.reis@mavt.ethz.ch (M.R.); xiaoxiang.yu@mavt.ethz.ch (X.Y.); nandan.maheshwari@mavt.ethz.ch (N.M.); fumiya.iida@mavt.ethz.ch (F.I.)

Mechanical Engineering Dept., Uludag University, 16059-Bursa, Turkey. E-mail: reis@uludag.edu.tr

AI Lab, Department of Informatics, University of Zurich, CH-8050 Zurich, Switzerland.