Spine morphology and energetics: how principles from nature apply to robotics

Being able to move in an energetically economical fashion is an important requirement for robots and animals alike [1–8]. In nature, energy in the form of food can be a scarce resource. Especially for cursorial animals, this creates a strong incentive to locomote in an efficient manner. Similar constraints apply to robots. Once they have been deployed for a mission in the field, autonomous robots typically do not have many opportunities to recharge their batteries. At the same time, such robots—and in particular legged robots—have only a limited capacity to carry batteries and other fuel sources. To achieve truly autonomous operation, conserving energy is imperative.

It has been shown that legged animals can achieve this economical motion partly by moving in ways that are effectively tuned to their natural mechanical dynamics [9–12]. That is, animals and humans take advantage of movement that arises through the passive interplay of gravity, ground contact forces, the elasticity in their muscles and tendons, and the mechanical inertia in their body segments. Using these natural dynamic motions allows them to store and recover mechanical energy and reduce the active work done by their muscles. Animals exploit, for example, the mechanical dynamics of an inverted pendulum as they walk with relatively straight legs [12]. When humans switch to running, they utilize the compliance in their tendons and muscles to store energy elastically while using a motion that exploits the mechanical dynamics of a spring loaded inverted pendulum [13]. Over the past years, these concepts have been steadily adapted into the world of legged robotics [14], and a number of robotic prototypes have been built that seek to take advantage of similar natural mechanical dynamics to achieve economical locomotion [15–19].

This exploitation of the natural mechanical dynamics has interesting implications for the design of robotic systems. The desire to move economically inherently couples the motion of a legged system to its morphology: the mechanical dynamics that a legged system exhibits are a direct consequence of the way it is built and these dynamics, in turn, dominate the way that an animal or robot should move in order to be economical. In this paper, we investigate a particular aspect of morphology and its potential usefulness in robotics: the articulation in the spine of quadrupeds.

In nature, the advantages of an articulated spine seem to be closely coupled to the choice of gait. At lower speeds, quadrupedal mammals generally utilize symmetrical gaits, in which the legs on the left side perform exactly the same motion as the legs on the right, just 180° out of phase. At higher speeds, they tend to transition to asymmetrical gaits, in which the two sides of the animal perform different motions and/or the phase shift differs from 180° [20]. Animals primarily utilize the articulation in their spine for these high speed asymmetrical gaits [21], whereas less spine motion can be observed in the symmetrical gaits [20]. This is an important observation and any investigation of the usefulness of an articulated spine in legged robots should thus incorporate the question of gait as a key factor in its analysis.

It has been argued that, with an asymmetrical gait, an articulated spine provides a number of dynamical benefits for animals with regard to leg recirculation, elastic energy storage, and stride length. In particular [22], pointed out that the articulation in the spine allows for faster recirculation of the legs. That is, the spinal rotation that is added in series to the flexion and extension of the hip and shoulder joints, enables a faster motion of the legs during swing. He further hypothesized that these additional degrees of freedom allow for longer stride lengths, attributing to a cheetah's high maximum speed [23]. Alexander [20] built upon Hildebrand's hypothesis when examining various models of quadrupedal animals and suggested that muscles and tendons in the articulation of the spine might act as additional elastic elements to store and release energy.

A number of studies have tried to assess the potential benefits of an articulated spine by performing comparative studies among quadrupedal models. Zhao et al [24], for example, developed a pneumatic-driven robotic quadruped with a rigid, passive, and active spine configuration. Using a step-function control pattern to achieve a bounding gait, they found that when the spinal motions are synchronized with leg movements, the extension and flexion of the active spine allows the robot to reach higher speeds. Kani and Ahmadabadi [25] used a central pattern generator controller to obtain bounding gaits for a robotic simulation comparing rigid, passive articulated, and active articulated spinal configurations. They found that bounding with a series elastic actuated spine performed better than an articulated passive spine and rigid spine in terms of bounding power consumption. Haueisen [26] compared two 2D quadrupedal bounding models, one with an articulated spine modeled as 6 rigid segments and one with a rigid spine modeled as 5 rigid segments. With these models, Haueisen analyzed the effects of speed and stride frequency on the energy requirements of both models for the bounding gait. She found that, during bounding, the articulated model utilized its articulated spinal joint in similar ways to that observed in nature and also found energetic benefits at higher speeds. Using optimal control Cao and Poulakakis [27], found that, at sufficiently high speeds, an articulated model of a bounding quadruped was more economical than a rigid model. All of these prior comparative simulation studies considered simplified quadrupedal models that represented a two-legged planar bounding robot. The comparison of the two morphologies has also been explored for bounding in hardware [28], where, in contrast to the above studies, they did not observe an energetic improvement for the flexible spine.

To extend these comparison studies beyond bounding, in this work, our model incorporates four independent legs. This design choice allows us to explore the effect of an articulated spine over the full range of quadrupedal gaits found in nature. This is an important extension, as bounding rarely appears as the gait of choice [29], while other gaits such as walking, trotting, tölting, and galloping are employed by a significant number of quadrupeds of all sizes [30]. Furthermore, our prior work has shown that bounding was never the most economical gait for robots [8]. In addition to the four legs, our model incorporates complexities such as feet with mass, inertia in the legs and torso, detailed motor models with realistic limitations on torque and speed, as well as series elastic springs with damping that generate the rich natural mechanical dynamics necessary for efficient locomotion. These complexities allow for physically realistic motions across a wide range of velocities and gaits.

Utilizing this model, in this paper we investigate whether and how the benefits of an articulated spine translate from nature into the world of robotics. We consider the choice of gait and investigate the three mechanical mechanisms stated above. Our tool of choice for this investigation is physics based simulation which we use to compare two model instances—one with and one without an articulated spine. The coupling of motion and morphology poses a particular challenge in this context: in order to optimally exploit their respective natural dynamics, each of the two morphologies will require distinctly different actuation patterns and joint movements to move forward in an economical fashion. Using the same control strategy for both morphologies would implicitly bias the comparison to favor one or the other. To overcome this issue, we let a numerical optimizer find the best possible joint trajectories, actuator inputs, and footfall timing to minimize the energetic cost of locomotion. By doing this individually for each morphology, we effectively remove the question of control from the comparison. This approach reduces the investigation of the benefits of an articulated spine solely to the mechanical differences between the two model instances. In the process, we create a causal relationship between morphology and performance. In addition to understanding the potential benefits of an articulated spine for robots, this approach also allows us to add new insight into the hypotheses put forth by Hildebrand and Alexander [20, 22, 23].

The dimensions and parameters of our model are roughly based off the quadrupedal robot, StarlETH [31, 32], and are similar to the models used in our previous studies on robotic gait selection [8, 33]. We created two model instances: one with a rigid main body and one with an articulated main body composed of two rigid segments connected by a rotational joint (figure 1). Similar to our previous work [8], we used optimal control to generate an energy cost landscape as a function of speed, using positive mechanical motor work normalized per distance traveled as the cost function. This was done for both model instances across a broad range of different gaits to allow for a fair comparison of the two morphological instances.

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In this study, we compare two different instances of a physics-based model of a quadrupedal legged robot: one with a rigid spine (figure 1(a)) and one with an articulated spine (figure 1(b)). We employ optimal control to identify the most economical motion for each of the two instances across a wide range of locomotion speeds and gaits, generating a lower bound on the energetic cost of locomotion. As the model and optimization methods used in this study are similar to the ones used in our previous work [8], we restate them only briefly while focusing on the aspects most critical to the comparison of the rigid versus articulated spine.

2.1. Model

The model is planar and consists of a main body and four individual legs with index $i\in \{LH, LF, RF, RH\}$ (labeled as left L, right R, hind H, and front F). Its geometrical dimensions are shown in figure 1. The individual segments of the model all have distributed masses with associated inertias. Each leg consists of an upper (m₂, j₂) and lower (m₃, j₃) segment that are connected with a prismatic joint that yields a variable leg length of l_i. The upper leg segments are attached to a main body at hip/shoulder joints (with joint angles $\alpha_{i}$) at a distance l_r from the center of mass (COM). The main body is either a single rigid segment with mass m_r and inertia j_r or—in the case of the articulated spine—consists of two identical segments of half the size with mass $m_{a}=\frac{1}{2}m_{r}$ and inertia $j_{a}=\frac{1}{8}j_{r}$. These two segments are connected with a rotational joint with joint angle θ. The COM of each segment is a distance l_a from this rotational joint. The position of the center of the main body is given by horizontal and vertical positions x and y, and by the orientation of the main body φ. For the articulated model, we use the orientation of the hind segment $\varphi_{H}$ for the same purpose.

There are three different joint types with index $p \in \{l, \alpha, \theta\}$ for leg extension, hip rotation, and spine rotation. All actuators are modelled as series elastic actuators (SEAs) [34], as they were used in our existing robotic hardware [35, 36]. Comparable to ligaments in nature, the series springs give the robot the opportunity to elastically store and release energy in all joints. Each joint is therefore modelled to consist of a spring with stiffness k_p and damping coefficient b_p that connects to the joint at its distal end while its proximal end is connected to an electrical DC motor and gearbox with reflected inertia $j_{{\rm ref}, p}$. For leg extension, this gearbox also converts the motor rotation into a linear motion.

For clarity, we will omit the indices i and p in the following description of the actuators. The position of an actuator (after the gearbox) is given by u and its speed by $\dot{u}$. The torque after the gearbox is given by T

In the SEAs, the joint torques/forces F are ultimately produced by the springs according to:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:SpringForces} {\bf F}_{l} = k_{l} \left(l_{o} + {\bf u}_{l} - {\bf l} \right) + b_{l} \left(\dot{{\bf u}}_{l} - \dot{{\bf l}} \right) \nonumber \end{align} \tag{ 1 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\bf F}_{\alpha} &= k_{\alpha} \left({\bf u}_{\alpha} - \boldsymbol{\alpha} \right) + b_{\alpha} \left(\dot{{\bf u}}_{\alpha} - \dot{\boldsymbol{\alpha}} \right) \nonumber \\ \nonumber F_{\theta} &= k_{\theta} \left(u_{\theta} - \theta \right) + b_{\theta} \left(\dot{u}_{\theta} - \dot{\theta} \right). \nonumber \end{align} \tag{ 2 }$$

With this, the generalized coordinate vector is given by ${\bf q} = \left(x , \; y , \; \varphi , \; \boldsymbol{\alpha} , \; {\bf l} , \; {\bf u}_{\alpha} , \; {\bf u}_{l} \right){\hspace{0pt}}^{T}$ for the rigid model and by ${\bf q} = \left(x , \; y , \; \varphi_{H} , \; \theta , \; \boldsymbol{\alpha} , \; {\bf l} , \; {\bf u}_{\alpha} , \; {\bf u}_{l} , \; u_{\theta} \right){\hspace{0pt}}^{T}$ for the articulated model. The vector of generalized torques is given by $\boldsymbol{\tau} = \left(0 , \; 0 , \; 0 , \; {\bf F}_{\alpha} , \; {\bf F}_{l} , {\bf T}_{\alpha} - {\bf F}_{\alpha} , \; {\bf T}_{l} - {\bf F}_{l} \right){\hspace{0pt}}^{T}$ and $\boldsymbol{\tau} = \left(0 , \; 0 , \; 0, \; F_{\theta} , \; {\bf F}_{\alpha} , \; {\bf F}_{l} , \; {\bf T}_{\alpha} - {\bf F}_{\alpha} , \; {\bf T}_{l} - {\bf F}_{l} , \; T_{\theta} - F_{\theta} \right){\hspace{0pt}}^{T}$, respectively.

All model parameters are provided in table 1. The parameters used, including mass, inertia, length, and spring stiffness, are based on our existing hardware [36]. This also holds for the motor/gearbox parameters, such as the reflected inertia $j_{{\rm ref}}$, the maximum output torque $T_{{\rm max}}$ limitation, and the speed limitation $\dot{u}_{\rm max}$. To reduce the number of free parameters, all values have been normalized with respect to total mass m_o, uncompressed leg length l_o, and gravity g.

Table 1. List of all model parameters. Values are expressed with respect to total mass m_o, leg length l_o, and gravity g.

Common properties:
m2  =  0.05 mo	m3  =  0.025 mo	j2  =  0.001 $m_{o}l_{o}^{2}$	j3  =  0.001 $m_{o}l_{o}^{2}$
bl  =  0.14 $m_o\sqrt{g/l_o}$	$j_{{\rm ref}, l}$  =  1.796 mo	$b_{\alpha}$  =  0.09 $m_o\sqrt{g/l_o}$	lh  =  0.75 lo
l2  =  0.25 lo	$j_{{\rm ref}, \alpha}$  =  0.449 $m_ol_o^2$	rfoot  =  0.05 lo	kl  =  5.0 $m_{o}g/l_{o}$
l3  =  0.25 lo	$k_{\alpha}$  =  2.5 $m_{o}gl_{o}/{\rm rad}$	$u_{{\rm max}, \alpha}$  =  0.79 ${\rm rad}$	$u_{{\rm max}, l}$  =  0.15 ${\rm l_o}$
$T_{{\rm max}, l}$  =  1.360 mog	$T_{{\rm max}, \alpha}$  =  0.680 $m_{o}g l_{o}$	$\dot{u}_{{\rm max}, \alpha}$  =  8.49 ${\rm rad}\sqrt{g/l_o}$	$\dot{u}_{{\rm max}, l}$  =  4.24 $\sqrt{gl_o}$
Rigid model:
mr  =  0.7 mo	jr  =  0.2 $m_{o}l_{o}^{2}$	lr  =  0.93
Articulated model:
ma  =  0.35 mo	ja  =  0.025 $m_{o}l_{o}^{2}$	$u_{\rm max, \theta}$  =  1.57 ${\rm rad}$	$k_{\theta}$  =  5 $m_{o}gl_{o}/{\rm rad}$
$T_{{\rm max}, \theta}$  =  0.680 $m_{o}g l_{o}$	$b_{\theta}$  =  0.24 $m_o\sqrt{g/l_o}$	$\dot{u}_{{\rm max}, \theta}$  =  8.49 ${\rm rad}\sqrt{g/l_o}$	la  =  0.5 lr
$j_{{\rm ref}, \theta}$  =  0.449 $m_ol_o^2$

The equations of motion (EOMs) for this model are stated in a floating base description using additional contact constraints (as described in detail in [14]). They are expressed as a differential algebraic equation:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:EOM} {\bf M} \left({\bf q} \right) {\ddot {\bf{q}}} = {\bf h} \left({\bf q}, {\dot {\bf{q}}} \right) + \boldsymbol{\tau} + {\bf J}_{c}^{T} \left({\bf q} \right) \boldsymbol{\lambda} \nonumber \end{align} \tag{ 3 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:DAE} \boldsymbol{\phi}_{c} \left({\bf q}\right) = {\bf 0}, \nonumber \end{align} \tag{ 4 }$$

with the mass matrix ${\bf M}$, the differentiable force vector ${\bf h}$, and the generalized torque vector $\boldsymbol{\tau}$. The algebraic function $\boldsymbol{\phi}_{c} \left({\bf q}\right)$ encodes the currently active contact constraints. It ensures that all the feet that are in contact with the ground do not slide or penetrate into the ground. The partial derivative of this function yields the contact Jacobian ${\bf J}_{c} = \partial{\boldsymbol{\phi}}_{c} / \partial{{\bf q}}$ whose transpose maps a vector of contact forces $\boldsymbol{\lambda}$ into the generalized coordinate space.

The sequence of footfalls that happens during locomotion breaks the simulation into individual phases (with index c) and each phase uses a different contact constraint function $\boldsymbol{\phi}_{c}$. At the end of each phase c (at time t_c) a specific number of feet must get in contact with the ground or leave the ground in order to transition into the next phase c  +  1. This requirement is encoded by an event function $\boldsymbol{\psi}_{c}$ that is evaluated at time t_c:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Event} \boldsymbol{\psi}_{c} \left(t_{c}\right)={\bf 0}. \nonumber \end{align} \tag{ 5 }$$

Furthermore, at the end of each phase, a collision might happen and velocities change instantaneously to match the new contact constraints $\boldsymbol{\phi}_{c+1}$. This instantaneous change in speed can be computed from:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:veldif} {\bf M}\left({\dot {\bf{q}}}^{+}\left(t_{c}\right) - {\dot {\bf{q}}}^{-}\left(t_{c}\right) \right) = {\bf J}_{c+1}^{T}\boldsymbol{\Lambda}_{c} \nonumber \end{align} \tag{ 6 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:velconst} {\bf J}_{c+1} {\dot {\bf{q}}}^{+}\left(t_{c}\right) = {\bf 0} \nonumber \end{align} \tag{ 7 }$$

with $\boldsymbol{\Lambda}_{c}$ being the vector of impulses associated with the collision. Matlab functions to calculate $\boldsymbol{\phi}_{c}$, ${\bf M}$, ${\bf J}_c$, and ${\bf h}$ from equations (3) and (4) are supplied in the supplemental files.

By prescribing a particular sequence of functions $\boldsymbol{\phi}_{c}$ and $\boldsymbol{\psi}_{c}$, we determine the gait of the model. The timing of events ${\bf t} \in \{t_{o}, , \cdots, \;t_{c}, \;\cdots, \;t_{\rm end}\}$ is left open. With this, we enforced eight different footfall patterns which can be broadly classified into a set of symmetrical and asymmetrical gaits (footfall patterns shown in figure 2). Symmetrical gaits are those 'in which the left and right feet of each pair have equal duty factors and relative phases differing by 0.5' [29]. Equivalently, for these gaits, we only simulate half the stride and mirror it to obtain the second half. Asymmetrical gaits are those that do not meet these criteria. The symmetrical gaits we investigate are four-beat walking, two-beat walking, tölting, and trotting. The asymmetrical gaits we investigate are bounding, grounded bounding, galloping, and gallop bounding. The majority of these gaits were chosen based off of our previous work [8]. Grounded bounding and gallop bounding were included because at high speeds, the optimizer chose phase durations within bounding and galloping that approached these gaits.

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2.2. Optimization

To compare the energetic economy of both model instances we considered the positive motor work cost of transport (COT). To this end, the positive motor work is computed as the integral of positive mechanical power summed over all motors over the duration of the stride and normalized by total mass and distance traveled:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:COT} {\rm COT} = \frac{\int_{t_0}^{t_{\rm end}} \sum_{i,p} \max \left(T_{i,p} \cdot \dot{u}_{i,p} , 0\right) {\rm d}t}{m_{o} g\left(x(t_{\rm end}) - x(t_0)\right)}. \nonumber \end{align} \tag{ 8 }$$

To improve the numerical behavior, we approximated the discontinuity in the cost function by smoothing the ${\rm max}({\rm value}, 0)$ function using the equation:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:CostSmoothing} {\rm max}(s,0) \approx \sigma {\rm log}\left(1+{\rm e}^{\frac{s}{\sigma}}\right) \nonumber \end{align} \tag{ 9 }$$

with $\sigma = 1 \times 10^{-5}$.

With this, we can set up the search for the optimal motion to be a constrained optimization problem, in which we identify optimal joint trajectories ${\bf q}(t)$, torque inputs ${\bf T}(t)$, and event times ${\bf t}$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:costfunction} \min \;\; {\rm COT} \left({\bf q}(t), {\bf T}(t), {\bf t}\right) \nonumber \end{align} \tag{ 10 }$$

subject to the following constraints:

• The trajectories ${\bf q}(t)$ and ${\bf T}(t)$ comply with the dynamics defined in equations (3) and (4).
• For each phase c, it holds:
  • $t_{c}>t_{c-1}$.
  • The event is triggered at the end of the phase according to equation (5).
  • Velocities $\dot{{\bf q}}$ change at the end of each phase as defined by equations (6) and (7).
• The motion is periodic; i.e. ${\bf q}(t_{o}) = {\bf q}(t_{\rm end})$ and $\dot{{\bf q}}(t_{o}) = \dot{{\bf q}}(t_{\rm end})$
• Motor torques, speed, and positions are within the given limits for all joints i and p:
  • $-T_{{\rm max}, p} < T_{i, p}(t) < T_{{\rm max}, p}$.
  • $-\dot{u}_{{\rm max}, p} < \dot{u}_{i, p}(t) < \dot{u}_{{\rm max}, p}$.
  • $-u_{{\rm max}, p} < u_{i, p}(t) < u_{{\rm max}, p}$.
• Vertical foot positions are non-negative, ${\bf y}_{\rm foot}\geqslant{\bf 0}$.

This constrained optimization problem was solved through a multiple shooting optimization framework (MUSCOD) (Diehl et al[37]) with methods illustrated and detailed in [8]. We examined locomotion velocities between 0.025 $\sqrt{l_{0}g}$ and 4.5 $\sqrt{l_{0}g}$. For each gait, we conducted an initial optimization at an initial speed and iteratively conducted optimizations at neighboring velocities in a branching method until the full range of velocities was covered or no solutions could be found. In this process, we used previously obtained solutions as the initial conditions of the neighboring velocities. At each particular speed, the solution with the lowest cost was taken as the optimal motion and re-used for re-initialization. This procedure was repeated from a variety of initial seeds to avoid local optima.

On average, the optimization of a single gait at a particular speed converged in about 5 minutes on one core of a 2.40 GHz processor. The time it took to process a given gait over the entire range of velocities varied largely. Depending on how well neighboring data points could be used for seeding new runs and on how many data points were recomputed, it took between 2 (trotting) and 28 (gallop-bounding) days to process a gait with the chosen speed resolution.

We found that the articulation in the spine has been used primarily in the asymmetrical gaits. The optimal solutions for the bounding and galloping gaits had large joint motion in the spine, while the spinal joint underwent relatively small deflections over the course of a stride for symmetrical gaits (table 2). An exemplary motion (galloping at $2.9\sqrt{l_og}$) is shown for both model instances in figure 3. More representative motions for each gait and model can be found in the supplementary videos (stacks.iop.org/BB/13/036002/mmedia). The resulting cost landscapes are shown in figure 4. Table 3 lists the minimal and maximal speeds for which a solution for each gait was found for each model instance, as well as the average changes in energetic cost and stride length between the two model instances. These averages were computed over the range of speeds at which solutions were found for both model instances and tested for statistical significance.

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Table 2. Change in spinal angle at a speed of 1.0$\sqrt{l_{o}g}$.

Gait	$\Delta \theta$
Four-beat walking	8.04 °
Two-beat walking	4.14 °
Tölting	7.83 °
Trotting	5.20 °
Bounding	51.5 °
Grounded bounding	56.4 °
Galloping	22.4 °
Gallop-bounding	112.7 °

Table 3. Speed range, difference in COT, and difference in stride-length for each investigated gait. p-values are based on a two-sample t-test.

Gait	Lowest speed$(\sqrt{l_og})$		Highest speed$(\sqrt{l_og})$		Mean COT difference		Mean stride length difference
	Min speed in search: 0.025		Max speed in search: 4.500
	Rigid	Articulated	Rigid	Articulated	From rigid	p-value	lo From Rigid	p-value
Four-beat walking	0.025	0.081	1.644	1.925	0.078	<0.001	0.257	<0.001
Two-beat walking	0.025	0.025	1.413	1.569	0.001	0.895	−0.007	0.862
Tölting	0.025	0.025	3.394	3.556	−0.012	0.320	0.157	0.072
Trotting	0.025	0.025	3.294	3.306	0.004	0.578	0.110	0.386
Bounding	0.031	0.025	2.994	4.500	−0.106	<0.001	0.464	<0.001
Grounded bounding	0.075	0.131	4.156	4.500	−0.080	<0.001	0.582	<0.001
Galloping	0.094	0.106	3.338	4.500	−0.034	<0.001	0.316	<0.001
Gallop bounding	0.106	0.338	3.469	4.500	−0.054	<0.001	1.413	<0.001

Among the symmetrical gaits, only four-beat walking showed a significant difference (p  <  0.05) between the rigid and articulated models in terms of energetics. Walking with an articulated spine increased energy consumption by 31.8%. All other symmetrical gaits had comparable energetic costs between the rigid and articulated models. Maximal speeds were constrained by the available torque and force capability in the motors. For all symmetrical gaits, the highest speed at which solutions were found was slightly higher for the model instance with the articulated spine.

The benefit of the articulated spine was clearly visible for the asymmetrical gaits, in which the articulation led to improved energetic economy and higher maximal speeds. Table 3 shows that the articulated model was significantly more economical than the rigid model for all asymmetrical gaits. Furthermore, for all asymmetrical gaits on the articulated model, we found solutions all the way to the upper bound of the speed range that we investigated (4.5 $\sqrt{l_og}$). For the rigid model, in contrast, the maximal speed was constrained to lower values (<4.2 $\sqrt{l_og}$) for all gaits, as the motors ran into torque limits. These torque limits were usually reached by the leg extension motors during the stance phase and by the hip motors during the swing phase preempting a particular leg's stance phase. Motor speed limits were generally not reached.

Confirming the results from our previous work [8], the combined cost landscapes showed that minimizing energy cost required switching from gait to gait as speed varies (figure 5). This choice of gait was slightly different for the two model instances. The rigid instance went from four-beat walking at low speeds, to trotting at intermediate speeds, to galloping at high speeds, to grounded bounding at the highest speeds. The articulated instance had the same gait sequence at low, intermediate, and high speeds, but transitioned to gallop bounding at the highest speeds. The biggest difference between the two model instances was observed for the transition speed between trotting and galloping. In particular, the results of the rigid (articulated) model instance indicated a transition from four-beat walking to trotting at a speed of 0.60$\sqrt{l_{o}g}$ (0.53$\sqrt{l_{o}g}$) and from trotting to galloping at a speed of 2.05$\sqrt{l_{o}g}$ (1.28$\sqrt{l_{o}g}$). The rigid instance transitioned from galloping to grounded bounding at a speed of 3.34$\sqrt{l_{o}g}$ and the articulated instance transitioned from galloping to gallop bounding at a speed of 3.63$\sqrt{l_{o}g}$.

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To further investigate the source of the energetic improvements, we focused on the energetic breakdown for galloping (figure 6). Similar results were observed for the other asymmetrical gaits. At a speed of 3.34$\sqrt{l_{o}g}$, the maximum speed at which we were able to find a rigid galloping solution, the rigid model had a mechanical COT of 0.390 and the articulated model had a COT of 0.284 (Note that the COT, which represents the energy consumed divided by the robot weight and distance traveled (section 2.2), is a unitless quantity). For both models, collisions were the smallest source of losses. At this speed, collision losses accounted for 0.041 of the losses for the rigid model, and for 0.034 of the losses for the articulated model. Damping losses contributed approximately equally to the COT for both models: 0.187 for the rigid model and 0.182 for the articulated model. With these losses being similar, the difference in energetic cost was thus driven primarily by negative motor work. Starting at a speed of around 1.5$\sqrt{l_{o}g}$, the negative motor work in the rigid model instance increased relative to the articulated model instance, coinciding directly with the worsened energetic economy of rigid galloping. At a speed of 3.34$\sqrt{l_{o}g}$, for the rigid model, negative work accounted for a partial COT of 0.162 and for the articulated model for a partial COT of 0.068.

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The primary source of this increased negative motor work were the hip motors used in leg recirculation (figure 7). At a speed of 3.34$\sqrt{l_{o}g}$, the hips performed 0.154 of the total negative motor work for the rigid model, while only performing 0.033 of the total negative motor work for the articulated model. At that same speed, the legs performed 0.008 of the total negative motor work for the rigid instance and 0.007 of the total negative motor work for the articulated instance. For the articulated model, the motor at the spine performed 0.028 of the total negative work.

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For asymmetrical gaits, the articulated model used the spinal joint to perform significant amounts of positive work. Table 4 shows the breakdown of positive work performed by the spinal joint for each asymmetrical gait at its maximum speed. It shows the total positive work done by the spinal joint, the portion done actively by the motor, the portion performed passively by the spring, and the portion done by the reflected motor inertia. Note that because the quadrupedal model is not energetically conservative, the value for the joint is less than the sum of the motor, spring, and inertia contributions. The spring performs a relatively small proportion of the positive work compared to the motor and inertia. It is not used to store large amounts of energy. This lack of energy storage is true across all speeds. Galloping is shown as an exemplary case in figure 8. The joint produces a much larger amount of positive work than it absorbs negative work.

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Among the symmetrical gaits, only four-beat walking showed a significant difference (p  <  0.05) between the rigid and articulated models in terms of stride length. Walking with an articulated spine increased step length by 10.4%. All other symmetrical gaits had comparable stride lengths between the rigid and articulated models. For all asymmetrical gaits the articulated model employed significantly longer stride lengths than the rigid model (table 3).

In this paper, we explored the energetic benefits of including an articulated spine in the torso of a quadrupedal robot. We compared the positive mechanical work COT for two model instances—one with and one without an articulated spine—across multiple gaits and a wide range of locomotion velocities. Using optimal control, we determined the best possible joint trajectories, actuator inputs, and footfall timing to minimize this COT individually for each gait, speed, and model instance. This was done to allow an adaptation of the motion to the respective morphology and to remove the question of control from the comparison. The model that we used built upon the large body of prior work comparing rigid and articulated spines by implementing a number of realistic characteristics, including series elastic actuation, an actuator model with limits on available torques and speeds, and most importantly, four independent legs that allowed us to explore the full range of quadrupedal gaits found in nature.

At higher locomotion velocities (>1.28$\sqrt{l_{o}g}$), our comparison revealed large energetic improvements for the articulated spine. In this speed range, asymmetrical gaits (in particular grounded bounding, galloping and gallop bounding) were the most efficient way of locomoting. For the four asymmetrical gaits that we explored, the average savings (within the speed range in which solutions were found for both model instances) were between 17.3% (galloping) and 38.0% (bounding). Furthermore, for the model with the articulated spine, we were able to find solutions at much higher speeds than for the model with the rigid spine, whose maximal speed was limited by the torque limits in the actuators. At lower speeds, symmetrical gaits, in particular four-beat walking and trotting, were optimal. In this speed range, the articulated spine had no positive effect. For symmetrical gaits, the cost values were nearly identical for all gaits. The only exception was four-beat walking with an articulated spine, which was, on average 31.8% more energetically costly than four-beat walking with a rigid spine. Comparing the most economical gaits at each speed, the articulated spine reduced the positive mechanical COT on average by $20.5{\%}$ across the entire range of velocities.

The benefits of the articulated spine have clearly been a function of the choice of gait. Improvements have primarily been observed for asymmetrical gaits. For symmetrical gaits, the energetic economy was similar for both models. In fact, these symmetrical gaits tend to not use the articulation in the spine and the angle of the spinal joint underwent relatively small changes over the course of a stride (table 2). This is not particularly surprising, since for the symmetrical gaits, the legs of a pair move in opposite directions. That is, any rotation of the spine would only aid one of these legs, while hindering the other. Consequently, the optimizer choose almost no motion in the spine for symmetrical gaits. The same reasoning was put forth by Alexander [20]. He stated that animal energy storage in the longissimus aponeurosis ligament of the spine would not be useful for symmetrical gaits, as in these gaits, 'the left leg of a pair swings forward while the right leg swings back'.

We further found that our robot model benefited from the same mechanistic effects that facilitate asymmetrical gaits in nature: the model exhibited improved leg recirculation, elastic energy storage in the spine, and enlarged stride lengths [20, 22, 23].

4.1. Leg recirculation

The articulated model's improved leg recirculation manifested itself in two ways: it increased the top speeds that the model was able to achieve (table 3) and it decreased the amount of negative work performed by the motors (figure 6). In the rigid model, the main body can only pitch in one direction. During phases of the gait in which the hind and front legs are moving in opposite directions (e.g. when the hind legs move backwards during stance to push off while the front legs move forwards to extend the stride and prepare for impact), the main body pitch can only aid one of these motions while impeding the other (figure 9). This leads to an increase in work that needs to be done by the hips, and eventually results in an increase in negative work that is done by the hip motors (figure 7). This negative work drives up the mechanical cost of transport for the rigid model. The decoupling of the front and back halves of the torso in the articulated model reduces the negative work done by the hip motors and minimizes the overall energetic cost. It further allowed the articulated model to move its legs quicker without violating torque constraints on the motors; resulting in higher maximal speeds than the rigid model instance. This argument builds upon the hypothesis by Hildebrand [22] about leg recirculation where he states that the flexible spine helps by 'advancing the limbs more rapidly'. Here it is clear that the articulated spine also helps by slowing the limbs without removing excess energy from the system.

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4.2. Elastic energy storage in the spine

4.3. Enlarged stride length

This material is based upon work supported by the National Science Foundation under Grant No. 1453346.