Pacific lamprey inspired climbing

The Newton–Euler method is used to derive the dynamic equations of motion and to calculate the internal forces required for robot design and dynamic equation transition. Equations for the ith link in series are given by equation (1)

$$\begin{align} \begin{aligned} Fx_i-Fx_{i+1}-\text{sgn}(\dot{x}_i)\,C_x &=m\,\ddot{x_i}\\ -m\,g\,\cos(\alpha)+Fy_i-Fy_{i+1}-\text{sgn}(\dot{y}_i)\,C_y &=m\,\ddot{y_i}\\ \frac{L}{2}\left(Fy_i+Fy_{i+1}-Fx_i-Fx_{i+1}\right)&\\ +\tau_{i-1}-\tau_{i} &= \frac{k_{\mathrm{isf}}}{12}m\,L^2\,\ddot{\theta}_i \end{aligned} \end{align} \tag{ 1 }$$

where Fx_i and Fy_i are the reaction forces from the revolute joints in the horizontal and vertical directions respectively, $k_{\mathrm{isf}}$ is a unitless scaling factor to account for gearbox inertia. C_x and C_y are environmental drag forces in x and y directions. Reaction forces and torques from the next distal link are absent for the last link and thus $Fx_{i+1}$, $Fy_{i+1}$ and τ_i are zero for the last link. For the stance phase the Cartesian positions ($x_{1-3}$, $y_{1-3}$) and speeds ($\dot{x}_{1-3}$, $\dot{y}_{1-3}$) of links are expressed in terms of the generalized coordinates and speeds of the system state $\theta_{1-3}$ and $\dot{\theta}_{1-3}$, respectively. Here link one is the head link and link three is the posterior link. The system of nine equations can be obtained for $\ddot{\theta}_{1-3},\,Fx_{1-3}$ and $Fy_{1-3}$ as a function of the system states ($\theta_{1-3}$ and $\dot{\theta}_{1-3}$) and input torques $\tau_{1-2}$. The final governing equation can be written as,

$$\begin{align} \mathbf{M}(\boldsymbol{\theta}) \begin{bmatrix} \ddot{\boldsymbol{\theta}}\\ \mathbf{f} \end{bmatrix} = \mathbf{C}(\boldsymbol{\theta})\,\dot{\boldsymbol{\theta}}^2 +\mathbf{G}(\boldsymbol{\theta},\dot{\boldsymbol{\theta}}) +\mathbf{U}\mathcal{T} \end{align} \tag{ 2 }$$

where

$$\begin{align*} \begin{array}{c c c c} \boldsymbol\theta = \begin{bmatrix} \theta_1 \\ \theta_2 \\ \theta_3 \end{bmatrix} & \dot{\boldsymbol{\theta}}^2 = \begin{bmatrix} {\dot \theta_1}^2\\ {\dot \theta_2}^2\\ {\dot \theta_3}^2\\ \end{bmatrix} & \mathbf{f} =\begin{bmatrix} Fx_{1}\\ Fy_{1}\\ Fx_{2}\\ Fy_{2}\\ Fx_{3}\\ Fy_{3}\\ \end{bmatrix} & \mathcal{T} =\begin{bmatrix} \tau_1\\ \tau_2\\ \end{bmatrix} \end{array} \end{align*}$$

while

$$\begin{align*} Fy_1 > 0. \end{align*}$$

Here, $\mathbf M$ is a $9 \times 9$ matrix as a function of the robot geometry. The matrix $\mathbf C$ is a $9 \times 3$ is a nonlinear function of position and accounts for the coriolis forces. The symbol $\dot{\boldsymbol \theta} ^2$ is a $3\times 1$ vector of the element-wise squares of the time derivatives of θ , and $\mathbf G$ is a $9\times1$ vector represents the effect of gravity and environmental forces acting on the linkages. U is a $3 \times 2$ input matrix with −1 on the main diagonal and 1 on the subdiagonal and zeros everywhere else. These equations apply while Fy₁ is positive.

During the flight phase, the head is free to translate as well as rotate, and the dynamics of the serial linkage can be described with five position states and five velocity states instead of the previous three and three, respectively. Similarly, four reaction forces are required instead of the previous six. The coupled ODEs with nine equations and nine unknowns can be expressed as:

$$\begin{align} \mathbf{M}(\boldsymbol{\theta}) \begin{bmatrix} \ddot{\boldsymbol{\theta}}\\ \ddot x\\ \ddot y\\ \mathbf{f} \end{bmatrix} = \mathbf{C}(\boldsymbol{\theta})\,\dot{\boldsymbol{\theta}}^2 +\mathbf{G}(\boldsymbol{\theta},\dot{\boldsymbol{\theta}},\dot x,\dot y) +\mathbf{U}\mathcal{T} \end{align} \tag{ 3 }$$

where $\mathbf{f} = \begin{bmatrix} Fx_{2}& Fy_{2}& Fx_{3}& Fy_{3} \end{bmatrix}^T$ and (x,y) is the Cartesian position of the first link. These equations apply while the head's vertical speed is positive in the inertial reference frame.

The conditions that result in transitions between the flight and stance phases are functions of the states. Our model assumes lift-off occurs when the y component of the head reaction force becomes zero.

When this occurs, the system states at the end of the stance phase are mapped into the initial states of the flight phase according to the following transform,

$$\begin{align} \begin{pmatrix} \theta_{1fi}\\ \dot{\theta}_{1fi}\\ \theta_{2fi}\\ \dot{\theta}_{2fi}\\ \theta_{3fi}\\ \dot{\theta}_{3fi}\\ x\\ \dot{x}\\ y\\ \dot{y}\\ \end{pmatrix} & = \begin{pmatrix} 1 & 0 & &\dots & 0\\ 0 &\ddots& & &\\ \vdots & & & & \vdots\\ & & & \ddots &0\\ 0 & \dots& & 0& 1\\ 0 & \dots & & &0\\ 0 & -\frac{L}{2}\,\text{s}(\theta_1) & 0 &\dots & 0\\ 0 & \dots& & & 0\\ 0 & \frac{L}{2}\,\text{c}(\theta_1) & 0 & \dots & 0\\ \end{pmatrix} \begin{pmatrix} \theta_{1s}\\ \dot{\theta}_{1s}\\ \theta_{2s}\\ \dot{\theta}_{2s}\\ \theta_{3s}\\ \dot{\theta}_{3s}\\ \end{pmatrix}\nonumber\\ & \quad + \begin{pmatrix} 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \frac{L}{2}\text{c}(\theta_1)\\ 0\\ \frac{L}{2}\text{s}(\theta_1)\\ 0\\ \end{pmatrix} \end{align} \tag{ 4 }$$

where ${\mathrm c}(\theta)$ and ${\mathrm s}(\theta)$ stand for cos and sin respectively. In equation (4) the vector on the left-hand side is the initial conditions for the flight phase states and the vector containing $\theta_{1-3}$ and $\dot{\theta}_{1-3}$ on the right-hand side is the states at the end of the stance phase.

We base our model on the passive attachment directional adhesion mechanism resembling geckos and cockroaches which achieve good performance in other dynamic climbing studies. Previous studies show that such a passive attachment can be modeled by assuming the negative spine velocity defines the moment of sticking [31]. As such, the exit condition for the flight phase is based on the relative velocity of the microspine array at the head of the serial linkage given by the following equation for the vertical velocity of the head:

$$\begin{align} \dot{y}_{\mathrm{head}} = \dot{y}+\frac{L}{2}\cos(\theta_1-\pi)\,\dot \theta_1 > 0. \end{align} \tag{ 5 }$$

In summary, climber dynamics are governed by equation (2) while the head force is positive; then, the stance states are mapped to the flight states by the transformation given in equation (4). Finally, the climber motion is governed by equation (3) until the vertical head velocity reduces to zero when the flight phase ends. The resulting hybrid dynamic motion governed by the stance and flight phases constitutes a single jump.

Hirose's early snake robot control policies prescribed feedforward or open-loop joint angle profiles generating snake-like motion he called serpanoid curves [32]. Later, Hatton and Choset [26] developed methods for generating joint space trajectories or gaits that generate desired locomotion shapes using 'keyframes'. Hatton's process produced the helix shape that enables tree climbing and other scripted gaits. Later improvements in this approach allowed for climbing on changing pole diameters and delicate structures [27]. In the dynamic climbing and running community, the earliest successes were demonstrated with a feedforward control policy [33, 34]. In this work, we first explore a gait with a feedforward control policy of the joints angle that mimics a lamprey's neural 'clock' [35]. Our control policy parameterizes Trident Climbers' shape changes by specifying a compression speed, extension speed, compression magnitude, and rest time. For the three-link model employed here, we fully specify a control policy by describing the commanded joint speed in compression $\omega_{\mathrm C}$, the commanded joint speed in extension $\omega_{\mathrm E}$, an amount of compression $\gamma_{\mathrm{max}}$ and a rest time $t_{\mathrm r}$. This study reduces the parameter space by setting the second joint angle to always be 180^∘ out of phase with the first.

Given the desired joint angle profile in figure 4, we use a proportional derivative feedback control law based on the actual relative joint angles and compute the joint actuation torques [$\tau_{c1}\, \tau_{c2}$] as

$$\begin{align} \begin{aligned} \begin{pmatrix} \tau_{c1}\\ \tau_{c2} \end{pmatrix} = & K_{\mathrm p} \begin{pmatrix} \phantom{-}\gamma_c-\gamma_1\\ -\gamma_c-\gamma_2 \end{pmatrix} + K_{\mathrm d}\, \begin{pmatrix} \phantom{-}\omega_{\mathrm{C/E}}-\dot{\gamma}_1\\ -\omega_{\mathrm{C/E}}-\dot{\gamma}_2 \end{pmatrix} \end{aligned}. \end{align} \tag{ 6 }$$

The calculated joint torques $\tau_{c1}$ and $\tau_{c2}$ are subjected to the following motor model that limits the commanded joint torque to a saturation limit given by the linear motor model,

$$\begin{align*} \kern2.5pt \tau_{\mathrm{max}} = \left\{\begin{array}{llcc} \frac{v}{v_m}\tau_{\mathrm{ST}} , & \hspace{-0mm} -\infty < \dot \gamma \leqslant 0\\ -\frac{\tau_{\mathrm{ST}}}{\omega_{\mathrm{NL}}}\,\dot \gamma+\frac{v}{v_m}\tau_{\mathrm{ST}}, & 0 < \dot \gamma \leqslant \frac{v}{v_m}\omega_{\mathrm{NL}}\\ 0, & \frac{v}{v_m}\omega_{\mathrm{NL}} < \dot \gamma < \infty \end{array}\right.\qquad\quad \end{align*}$$

$$\begin{align*} \tau_{\mathrm{min}} = \left\{\begin{array}{llcc} 0, & -\infty < \dot \gamma \leqslant -\frac{v}{v_m}\omega_{\mathrm{NL}}\\ -\frac{\tau_{\mathrm{ST}}}{\omega_{\mathrm{NL}}}\,\dot \gamma-\frac{v}{v_m}\tau_{\mathrm{ST}} , & -\frac{v}{v_m}\omega_{\mathrm{NL}} < \dot \gamma \leqslant 0\\ -\frac{v}{v_m}\tau_{\mathrm{ST}}, & \hspace{0mm} 0 < \dot \gamma < \infty \end{array}\right. \end{align*}$$

where the parameters of the motor model are given in table 2. Other motor and control policy parameters were discovered through the numerical optimization in section 3. Having layed out our reduced order model for lamprey climbing we turn our attention to exploring the model numerically to gain design and locomotion insights.

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Preliminary testing revealed that drag plays a significant role in the dynamics of this form of climbing at this mass scale. In order to account for drag and spine release cost, Trident Climber was placed on its climbing surface and a pulley system was used to add weight incrementally to determine drag force in the y direction. The value of this drag force is reported on a per-link basis in table 2 as C_y . Lateral drag was similarly measured and reported in table 2 as C_x . Here we assume that lateral motion in each link's local coordinate frame, can be reasonably mapped to the global x direction and local link vertical motion is primarily in the y. In table 2 it can be seen that C_y is much greater than C_x , 0.48 vs 0.08 respectively. This disparity is due to the force required to make the microspines release. This spine release cost is not present for motion in the lateral direction. Moreover, the scaling factor $k_{\mathrm{isf}}$ to account for the gearbox reflected inertia was found to be 2.0.

In this study, we consider climbing behaviors that allow vertical and lateral motion. We quantify the performance of the climbing by evaluating the net vertical displacement, the net lateral displacement, and energetic efficiency. For each experiment, the robot was commanded to jump 29 times, 15 right and 14 left while a Vicon motion capture system with four calibrated cameras recorded Head position and orientation data at 100 Hz. Figure 5 shows reflective markers for the Vicon motion capture system. Experimental data of each jump was post-processed such that the vertical and lateral displacement was evaluated separately. The time-series data from the Vicon and the onboard joint encoder and current log were binned such that the mean and standard deviation of the peak jump could be determined for each experiment. Sample experimental results of this internal shape data for a 29 jump experiment are compared to the three-link model numerical results in figure 6. Here we see the correspondence between the simulation shape variables ($\gamma_1, \gamma_2$) and the shape variables as measured by Trident's incremental encoders. Figure 6 also depicts the simulated transition from stance to flight phases, shown with the dashed and dotted-dashed lines, respectively. An example of global reference frame data from the Vicon system is shown in figure 7. This plot gives the time histories for Trident's binned joint angles during one experiment.

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Efficiency is computed by comparing the ratio of work done by the motors at the output shaft to the increase in potential energy of Trident Climber. The work done by the motor is calculated by time integrating the instantaneous power of each motor. This is done using the measurements from the joint encoder and motor current according to equation (8)

$$\begin{align} \eta= \frac{m\,g\,\Delta y}{\int_0^{t_{\mathrm{flight}}}P_{\mathrm{motor}}\,{\mathrm d}t}= \frac{m\,g\,\Delta y}{k_{\mathrm T}\int_0^{t_{\mathrm{flight}}}\dot{\gamma}_1\,i_1 +\dot{\gamma}_2\,i_2 \,{\mathrm d}t} \end{align} \tag{ 8 }$$

where $\dot \gamma$ is the motor speed and $k_{\mathrm T}$ is the motor torque constant.

The outside columns of figure 8 show comparisons of the experimental data with error bars reflecting one standard deviation of data as in figure 7. For instance, the data represented in figure 7 is a single data point in the cyan trace at $\gamma_{\mathrm{max}} = 60^{\circ}$ in subfigure 8(c).

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Figure 8(a) shows the effect of compression speed on vertical displacement and each data set (single color) is a horizontal cross-section of figure 8(b). The jump height is positively correlated with the compression speed until around $\omega_{\mathrm C} \approx 1000\,\mathrm{deg/s}$, when it begins to decrease with increasing ω_c . Experimental observations and simulation results both display the same location of peak jump height and a slight dependence on the max joint angle $\gamma_{\mathrm{max}}$. However, the simulation model overestimates the jump height by about three millimeters.

Figure 8(c) shows the effect of the maximum commanded joint angle on the vertical displacement of a single jump and each curve is a vertical cross-section of figure 8(b). We observe that within the joint angle constraint of Trident Climber, vertical displacement is positively and linearly correlated with the max joint angle. Similar to what is observed in figure 8(a) the model over-predicts jump height by about three millimeters for most cases, except for low compression speeds. Peak vertical climbing displacement of 47 mm ±5 occurs for a commanded compression speed of ≈ 1000 deg s⁻¹ and $\gamma_{\mathrm{max}}$ of 90^∘.

Figures 8(d)–(f) shows the lateral displacement of a single jump. Like vertical jump displacement, there is a peak compression speed, ≈ 1000 deg s⁻¹, with slight dependence on joint angle. Also, the lateral displacement has a positive linear correlation with joint angle. Unlike vertical displacement, the model mostly underpredicts the lateral distance by about 10 to 15 mm.

Efficiency is shown in figures 8(g)–(i). In figure 8(h) we observe that a peak efficiency of $14\%$ occurs at $\gamma_{\mathrm{max}} = 90^{\circ}$ and $\omega_{\mathrm C} = 1000$ deg s⁻¹. Figure 8(i) reports efficiency based on the $\Delta y_{\mathrm{max}}$ (see figure 7). As such, this number is the same output as the Numerical study. In order to make a fair comparison of Trident's single jump data to continuous climbing and to other robots, we consider the net vertical displacement (see figure 7). Trident Climber slips some distance, typically about 5 mm, as its microspines engage the carpet. Consequently, the $14\%$ reported is the max efficiency that can be attained if the spines attach immediately at the apex. As such, the net efficiency for the peak efficiency operating point is $12\%$ or a specific resistance of 8.3.

After determining the control parameter space that relates to climbing performance, we consider steering in the context of vertical climbing. Figure 9 is generated by computing the vertical-to-lateral displacement ratio for the parameter variational study. This plot reveals how the actuation parameters ($\gamma_{\mathrm{max}}$, $\omega_{\mathrm C}$) affect steering on Trident. The actuation in the stared regimes can be used to steer Trident Climber, resulting in net motion that is either vertical or mostly lateral. The yellow contour labeled 'Vertical Regime' represents strides that have the largest vertical component. So if vertical progress is desired and high lateral deviation is acceptable, the 'Vertical Regime' is the optimal operating point. This vertical jumping regime is characterized by a large joint angle of 90^∘, and the commanded compression speed is appropriately matched to the characteristics of the motor (≈ 1000 deg s⁻¹ for our motor). To achieve mostly net vertical locomotion, the actuation ($\omega_{\mathrm C}$) must alternate sign since consecutive steps alternate displacement in the lateral direction. As such, two jump steps form a stride with a y displacement of two times the vertical step displacements reported in figure 8.

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If, on the other hand, the objective is to achieve lateral motion, then Trident can be actuated in the region labeled as 'Lateral Regime Left/Right.' Here the jumping is characterized by a low joint angle and a compression speed slightly faster than optimal for vertical climbing. For the marked case, the compression speed is ≈ 2000 deg s⁻¹. Considering that the entire blue contour with a ratio between 0.3 and 0.4 has a similar displacement ratio, priority is given to the higher compression speed since it results in faster cycle speeds. However, considerations other than speed might lead one to choose the regime on the left side of the blue contour. As shown in figure 8(c), the peak vertical displacement for this particular case $(\omega_{\mathrm C} = 2000$ deg s⁻¹, $\gamma_{\mathrm{max}} = 50^{\circ})$ is about 10 mm, accounting for spine engagement displacement cost results in close to zero net lateral displacement and 25 mm lateral jumps.

Another key result is that the body shape in the stance phase predicts the jump direction. By the convention in figure 3, positive compression speeds on the first motor $\dot{\gamma}_1$ result in shapes where the first two links are deflected with their convex sides pointing left and as a result, the net lateral motion is to the left. For example, the jumper in figure 3 is poised to jump left. Conversely, negative compression speeds, result in rightward lateral jumps.

By incorporating insights about locomotion direction and the ratio of vertical to lateral displacement into our feedforward control scheme, we were able to pick a proper pulse profile, similar to the one depicted in figure 9(b) to steer Trident Climber. Note that vertical locomotion utilizes a pulse with compression speed of alternating sign. The lateral motion utilizes small, fast pulses with a compression speed corresponding to the desired lateral displacement figure 10 shows Trident Climber demonstrating the utility of these actuation pulses for steering around an obstacle.

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The principle of impulse and momentum explains the positive correlation between max joint angle, vertical jump, and lateral motion. Impulse and momentum theory states that the change in the vertical velocity of Trident is proportional to the integral of the vertical head reaction force with respect to time

$$\begin{align} \Delta\,\vec{V} \propto \int \vec{F}(t) \,{\mathrm d}t. \end{align} \tag{ 9 }$$

Athletic jumps thoroughly saturate the motor and limit the head reaction force. Considering equation (9) and noting that though the value of $\vec{F}(t)$ is invariant due to saturation at any t, increasing the duration of the force profile will still increase the impulse and the vertical velocity. Conversely, when commanded speeds are outside the motor's limit, the head reaction force has a much shorter time interval to develop a vertical impulse. For this reason, peak performance observed in terms of height, efficiency, and to a lesser degree, lateral travel is due to a good matching of command pulse to motor power output and link length. Furthermore if maximizing vertical step displacement is the primary goal, consideration should be given to maximizing the range of joint motion.

The experimental results and the model predictions agree well except for at low compression speeds. The poor correlation between the experiments and the model in this region is presumably because the energetic cost of detachment is significant compared to the kinetic energy gained during the stance phase; therefore, the revolute joint assumption used to model the head attachment does not represent the dynamics of these cases.

Generally speaking, the stance-to-flight transition occurs when commanded compression ends and the commanded extension begins. In other words, the stance phase is roughly equivalent to when the body is compressing and a flight phase begins when the actual max joint angle is reached and the extension phase begins.

Kemp and Mosser documented that L. tridentata climbs with a mean stride length of 3.59 cm [28]. Trident Climber achieves a step height of $4.1\pm 0.25$ cm in the vertical climbing regime, about $14\%$ better than the mean cycle step height published for L. tridentata.

Considering figure 2, and applying the law of cosines to the triangles generated by shortening the overall length of the three links, we can write down the relationship between sinuosity Ω and $\gamma_{\mathrm{max}}$ given as

$$\begin{align} \gamma_{\mathrm{max}}(\Omega)=\pi-\arccos\left\{\frac{5}{4}-\frac{9\Omega^2}{4}\right\} . \end{align} \tag{ 10 }$$

Kemp et al publish that L. tridentata climb with a sinuosity of 88%. Equation (10) gives $\gamma_{\mathrm{max}}(0.88) \approx 60^{\circ}$ suggesting that Trident Climber's compression is comparable to observed compression in L. tridentata. Previously, Zhu and Mosser suggested that lampreys climb in such a way as to optimize their climbing efficiency. As can be seen in figure 8(h), $\gamma_{\mathrm{max}} = 60^{\circ}$ is indeed close to an efficiency peak for Trident Climber.

When we compare our compression and extension speeds of Trident Climber with previously published data from L. tridentata, Kemp et al [28] show the compression time of L. tridentata is about 160 ms, and this range is similar when looking at other videos of Lampreys climbing. Keeping the Froude number constant and scaling to Trident Climbers length scale, this corresponds to 450 deg s⁻¹ compression speed [38]. Figure 8(h) shows the efficiency of this location ($\gamma_{\mathrm{max}} = 60^{\circ}, \omega_{\mathrm C} = 450$ deg s⁻¹) is approximately 10%. The efficiency of this operating point is within 25% of the model's peak. This minor discrepancy is perhaps due to L. tridentata being force limited in its stance phase, muscles or oral disc holding power, or attachment dynamics.

We could not compare the lateral displacement performance between Trident Climber and its biological counterpart as, to our knowledge, no prior work has quantified the lateral motion by L. tridentata while climbing.