2.1. Basic operating principle

Design and motivation behind our spring-assisted MRR, including control framework based on joint torque sensing, was reported previously by Liu et al. [Reference Liu, Liu and Goldenberg4]. Each SA-MRR joint module constitutes a clutched PEA system (Fig. 2) with two mechanical operating modes: (Mode 1), “moving-freely mode,” and (Mode 2), “spring-connected mode” by convention [Reference Plooij, Wolfslag and Wisse6]. Instead of immobilizing the module, activating the electromechanical brake engages the embedded flat spiral torsion spring in parallel with the DC motor shaft and transitions the SA-MRR module from (Mode 1) to (Mode 2). Deactivating the brake disengages the spring and transitions the module from (Mode 2) to (Mode 1). The mechanical components unique to the spring-assisted MRR module contribute less than $ \text{1}$ % additional weight compared with our conventional MRR modules. Accordingly, added weight is not a significant concern when considering the energy consumption of SA-MRR modules.

Figure 2. Schematic diagram of SA-MRR joint module. For true joint module section views and actual photographs, the reader is referred to [Reference Liu, Liu and Goldenberg4].

2.2. Hybrid active–passive redundant actuation

Engaging the spring in parallel creates a hybrid active–passive redundant actuation scenario where the spring contributes to actuation torque and energy delivery. Springs are passive mechanical elements that may only recycle prestored energy and cannot arbitrarily incite actuation energy. For each joint operating in (Mode 2), spring-assisted mode, torque and stored energy of the embedded spring are position-dependent and proportional to the joint displacement angle and to its square, respectively. Behind each MWM intent is the understanding that spring torque $ \tau _{k}$ is equally capable of assisting or resisting instantaneous motor torque $ \tau _{m}$ . As such, each MWM strategy must generally be integrated into preplanned tasks. Meanwhile, (Mode 1) may comply globally with power-and-force limitations, as well with human–robot collaboration safety constraints for comprehensive SA-MRR safety in human environments.

2.3. SA-MRR system modeling

The spring at joint module $ i$ , ( $ i=1,\ldots,n$ ) co-operates in actuator space. In (Mode 1), joint torque $ \tau _{i}$ and motor torque $ \tau _{m_{i}}$ are related through the speed reducer ratio $ \gamma _{i}\gt 1$

(1) \begin{equation} \tau _{m_{i}} =\frac{\tau _{i}}{\gamma _{i}}. \end{equation}

Connecting the spring thus augments the required motor torque

(2) \begin{equation} \tau^{\prime}_{m_{i}} =\frac{\tau _{i}}{\gamma _{i}}-\tau _{k_{i}} \end{equation}

with spring torque, where prime notation denotes (Mode 2) operation. At module $ i$ , spring torque $ \tau _{k_{i}}$ depends firstly on the operating mode defined by the brake state $ b_{i}\in \mathbb{B}$

(3) \begin{equation} \tau _{k_{i}} =\begin{cases} \begin{array}{c@{\quad}l} \tau _{k_{i}}(\theta ), & \text{ brake engaged, $ b_{i}=1 $}\\[4pt] 0, & \text{ brake disengaged, $ b_{i}=0 $}\\ \end{array} \end{cases} \end{equation}

and secondly on its torque–deflection characteristic as a function of the spring deflection angle $ \theta$ . Linear torsion springs with rest angle $ \theta _{0}$ and positive stiffness $ k_{s}$

(4) \begin{equation} \tau _{k_{i}}(\theta ) =-k_{s_{i}}\Delta \theta _{i} =-k_{s_{i}}\left (\theta _{i}-\theta _{0_{i}}\right ) \end{equation}

are assumed for modeling the spring features. Subscripts $ i$ are often dropped hereafter for readability.

The $ n$ -DoF serial SA-MRR manipulator dynamics

(5) \begin{equation} M(q)\ddot{q}+C(q,\dot{q})\dot{q}+g(q)=\tau \end{equation}

has desired redundantly actuated torque $ \tau$ expressed

(6) \begin{equation} \tau =\gamma ({\tau^{\prime}_{m}}+\tau _{k}) \end{equation}

where $ \gamma =\mathrm{diag}(\gamma _{1},\ldots,\gamma _{n})$ . No spring augmentation design strategy of one joint affects any other SA-MRR module.

2.4. SA-MRR energy consumption measures

Design, optimization, and evaluation of each of the two energy-aware MWM strategies rely integrally on the following two energy consumption measures for the SA-MRR joint modules:

1. 1. The (Mode 1) motor energy. This is the active motor energy consumed per cycle by the desired repetitive task motion. It is also the initial motor energy expended on startup to set the spring connection angle:

   (7) \begin{equation} E_{m} =\int P_{m}(t)\text{d}t =\int _{t_{0}}^{t_{1}}\left (\frac{\tau _{m}(t)^{2}R_{m}}{k_{\tau }^{2}}+\left |\gamma \tau _{m}(t)\dot{q}(t)\right |\right )\text{d}t \end{equation}
   on $ t=[t_{0},t_{1}]$ , where $ P_{m}$ , $ R_{m}$ , $ k_{\tau }$ are the motor power, terminal resistance, and torque constant, respectively.

2. 2. The (Mode 2) motor energy. This is the active motor energy consumed per cycle while the spring is connected:

   (8) \begin{equation} {E^{\prime}_{m}} =\int{P^{\prime}_{m}}(t)\text{d}t =\int _{t_{0}}^{t_{1}}\left (\frac{{\tau^{\prime}_{m}}(t)^{2}R_{m}}{k_{\tau }^{2}}+\left |\gamma{\tau^{\prime}_{m}}(t)\dot{q}(t)\right |\right )\text{d}t \end{equation}
   where the variables common to (7) are the same as above.

These two measures, (7), (8), are also used to report simulation results in Section 4 to quantify energy effectiveness of both of the two proposed MWM strategies.

2.5. SA-MRR system operating assumptions

Every MWM method changes task-specific motor demands and SA-MRR module operating conditions. While complementary methods lessen the motor load, antagonistic methods add spring torque and may nearly overload the motor. In this work, we assume a simplified DC motor model with linear relation between torque and current throughout its entire operating range, including at a suitably high torque threshold $ |\tau _{m_{\text{max}}}|$ that is very near but sufficiently less than the stall torque. We also assume symmetry $ |\tau _{m_{\text{max}}}|=|-\tau _{m_{\text{max}}}|$ and motor ability to operate perhaps near-continuously around this threshold without performance degradation or heat damage.

While spring stiffness $ k_{s}$ is selected during design and is constant, it is fully reconfigurable offline through component swapping. For all SA-MRR operations in this research work, we assume the spring has no hysteretic effects, that it engages always from its undeflected equilibrium angle and that it influences the motor in no way besides its torque contribution. For the antagonistic method, the brake is required to operate during collaboration scenarios when the joint velocities are relatively lower, and we assume each joint brake changes state instantaneously with no position error.

2.6. SA-MRR segregation as primary safety intent

Segregation is unequivocally the highest, most reliable form of robot safety. Spatial segregation is upheld intuitively for collaborators when robots keep within an appointed territory. State-of-the-art robot safety may allow humans and robots within mutual reach, but under highly regulated conditions that often compromise task efficiency. Instead, the SA-MRR may allow uninterrupted robot operation by spatially segregating its reach from humans [Reference Patel, Singh and Liu28]. The SA-MRR safety technique we develop in this work begins by already knowing a task-specific set of joint angle extrema (endpoints) that have been chosen independently for segregation safety and should be safeguarded. Also in this research work, the SA-MRR collaboration intent is that of humans and robots performing separate tasks in relatively close proximity, and with relatively lower robot velocities, but not of hands-on-robot or direct physical interactions. The workspace of the robot is limited by the ranges of the joint springs, leaving more workspace for humans.

3.1. Spring assistance to safeguard endpoints

The first MWM strategy uses the spring to help achieve decisive, mechanical segregation of the robot arm and end-effector from the collaborators by safeguarding task-specific joint limits. While a spring is not a limiting device, we propose that a spring may be paired with a suitable task to resemble joint limiting with added safeguarding by the following rationale and method. If the available motor torque can be critically diminished using the spring, then the task endpoints may be held statically but not easily surpassed. Only retreat motion from the endpoints is possible in this static sense, and the approach toward endpoints is buffered safely by the spring. Thus, spatial segregation is achieved by mechanical limitation in addition to the task-specific software joint limits. However, not every task may allow for SA-MRR joint limit safeguarding, as described hereafter.

To implement this torque-based endpoint limiting strategy, the augmented motor torque ${\tau^{\prime}_{m}}(\theta )$ at each joint shall satisfy

(9) \begin{equation} {\tau^{\prime}_{m}}(\theta _{\text{min}})=-|\tau _{m_{\text{max}}}|, \end{equation}

(10) \begin{equation} {\tau^{\prime}_{m}}(\theta _{\text{max}})=+|\tau _{m_{\text{max}}}|, \end{equation}

and act progressively via spring torque antagonism.

The spring will be connected while approaching the limit angles and disconnected otherwise. This implies a position schedule of brake operation angles $ \theta _{b_{\text{min}}}$ , $ \theta _{b_{\text{max}}}$ between the endpoints

(11) \begin{equation} \theta _{\text{min}}\lt \theta _{b_{\text{min}}}\leq \theta _{b_{\text{max}}}\lt \theta _{\text{max}}. \end{equation}

The brake state $ b\in \mathbb{B}$ must change whenever either $ \theta _{b_{\text{min}}}$ or $ \theta _{b_{\text{max}}}$ is encountered, in either joint rotation sense, which is typically four times per cycle.

To satisfy ${\tau^{\prime}_{m}}(\theta _{\text{min}})$ and ${\tau^{\prime}_{m}}(\theta _{\text{max}})$ thresholds, (9)–(10) requires antagonistic spring torque (see Fig. 3) with endpoint magnitudes defined by the task, motor, and spring parameters

(12) \begin{equation} \tau _{k}(\theta _{\text{min}}) =-k_{s}\left (\theta _{\text{min}}-\theta _{b_{\text{min}}}\right ) =\tau _{m}(\theta _{\text{min}})+|\tau _{m_{\text{max}}}|, \end{equation}

(13) \begin{equation} \tau _{k}(\theta _{\text{max}}) =-k_{s}\left (\theta _{\text{max}}-\theta _{b_{\text{max}}}\right ) =\tau _{m}(\theta _{\text{max}})-|\tau _{m_{\text{max}}}|. \end{equation}

Selecting spring stiffness $ k_{s}$ as design parameter fully defines both brake operation angles $ \theta _{b_{\text{min}}}$ and $ \theta _{b_{\text{max}}}$ for a task

(14) \begin{equation} \theta _{b_{\text{min}}} =\theta _{\text{min}}+\frac{\tau _{m}(\theta _{\text{min}})+|\tau _{m_{\text{max}}}|}{k_{s}}, \end{equation}

(15) \begin{equation} \theta _{b_{\text{max}}} =\theta _{\text{max}}+\frac{\tau _{m}(\theta _{\text{max}})-|\tau _{m_{\text{max}}}|}{k_{s}}. \end{equation}

Furthermore, the task itself places an ultimate lower bound

(16) \begin{equation} k_{s} \geq \frac{\tau _{m}(\theta _{\text{min}})-\tau _{m}(\theta _{\text{max}})+2|\tau _{m_{\text{max}}}|}{\theta _{\text{max}}-\theta _{\text{min}}} \end{equation}

while theoretically $ k_{s}$ has no task-specific upper bound.

Figure 3. Waist joint torque–angle profiles for 3-DoF SA-MRR task with endpoint antagonism $ \tau _{k}(\theta _{\text{min}})$ and $ \tau _{k}(\theta _{\text{max}})$ changes ${\tau^{\prime}_{m}}(\theta )$ only on $ [\theta _{\text{min}},\theta _{b_{\text{min}}}]$ and $ [\theta _{b_{\text{max}}},\theta _{\text{max}}]$ . (a) Motor-only torque. (b) Spring torque underlaying spring-augmented motor torque. Fictitious $ \pm |\tau _{m_{\text{max}}}|$ is shown near for detail clarity.

For every potential selection of $ k_{s}$ and its prescribed brake operation cycle (angles), physical feasibility must be verified by ensuring $ |{\tau^{\prime}_{m}}(\theta )|=|\tau _{m_{\text{max}}}|$ only at $ \theta _{\text{min}}$ and $ \theta _{\text{max}}$ and that $ |{\tau^{\prime}_{m}}(\theta )|$ is sufficiently less than $ |\tau _{m_{\text{max}}}|$ everywhere else.

Features of the task-specific torque–angle profile may disqualify a candidate task from straightforwardly applying this antagonistic safety method to safeguard task-specific endpoint limit angles (Fig. 4). The first and most basic requirement of this method is for a unique torque value to exist at each endpoint limit angle, $ \theta _{\text{min}}$ , $ \theta _{\text{max}}$ . Neither standing still at endpoints nor precisely repeating a task will visibly change its torque–angle profile, and so both may be acceptable task features of this method. However, two task features which generally result in nonunique torque include adding or removing payload at either endpoint, and also if other robot joints have nonzero velocity while one joint is stationary at a limit. This list of task conditions and exclusions has not been exhaustively verified, but adherence to it should enable straightforward spring design and limit safeguarding no matter the SA-MRR structural configuration. Indeed, this list excludes many tasks from applying the MWM safeguarding method.

Figure 4. Flowchart illustration of the proposed endpoint safeguarding method.

Energy consumption properties of this MWM safety method as illustrated in Fig. 4 are related strongly to torque characteristics of the task, motor, and spring. It should be noticed that the primary intent of this safety method is not to save energy. In fact, only some tasks and some design conditions can save energy as a byproduct of this method. Poor design can lead to energy increase for every task, even if the task is one which may allow energy savings by appropriate energy-aware design. Since $ |{\tau^{\prime}_{m}}(\theta )|\gt |{\tau _{m}}(\theta )|$ at $ \theta _{\text{min}}$ and $ \theta _{\text{max}}$ irrespective of $ k_{s}$ , it is true for every SA-MRR joint that the active motor energy to stand still at endpoints is always higher with this safeguarding strategy than without it. While joints are in motion, the cumulative motor energy increases or decreases over the time domain movements on $ [\theta _{\text{min}},\theta _{b_{\text{min}}}]$ and $ [\theta _{b_{\text{max}}},\theta _{\text{max}}]$ . The extent of energy losses or savings for every task is known only by (8) after spring augmentation design and is impossible to judge from inspecting the torque–angle profile alone. In favorable tasks, energy savings may be realized by the following guidelines for spring design and SA-MRR module selection. When task-specific energy savings are possible, it is usually greatest when $ |\tau _{m_{\text{max}}}|$ minimally exceeds $ \text{max} \left \{|\tau _{m}(\theta )|\right \}$ . This suggests choosing the smallest SA-MRR joint module and actuator possible for the task if energy is a large concern. Also when energy savings is possible, ( ${E^{\prime}_{m}} \lt E_{m}$ ), the energy-optimal $ k_{s}$ is often identically the lowest by (16), or the smallest feasible stiffness exceeding it. Logically, a spring stiffness also exists for which endpoint safeguarding is theoretically net energetically neutral ( ${E^{\prime}_{m}} =E_{m}$ ), but feasibility is not guaranteed in general and must be tested. For tasks where energy necessarily increases ( ${E^{\prime}_{m}} \gt E_{m}$ ), ${E^{\prime}_{m}}$ decreases to a task-specific asymptotic minimum as the design specification of $ k_{s}$ approaches infinity (for all actuator sizes).

Nearly any spring with satisfactory $ k_{s}$ may be used for endpoint limit safeguarding. A spring already installed in any SA-MRR joint module is suitable for any variety of tasks for which the method of Fig. 4 flows directly from start to end terminal nodes without looping. Safety in many tasks is a higher priority than energy optimality. As such, the SA-MRR may seamlessly apply this MWM safety strategy to multiple tasks simply by reconfiguring the brake operation schedule of (11) online for each new task definition. In this way, the spring installed in each SA-MRR joint may be used across several safe manipulation task scenarios or used for other applications such as the following energy-saving MWM strategy.

3.2. Spring assistance for energy efficiency

The second spring-assisted MWM strategy is proposed to minimize motor energy per cycle during tasks confined for safety to small task-specific regions of the SA-MRR work envelope. The objective is to augment each task-specific torque–angle profile with the most energy-complementary spring characteristic by optimization methods, while also considering task-specific spatial safety constraints.

Augmenting motor torque with the torque characteristic of a linear spring allows design optimization of up to two parameters: stiffness $ k_{s}$ and rest angle $ \theta _{0}$ . Intuitively, $ \tau _{k}$ contributes to energy savings in confined regions largely by compensating gravity. Spring stiffness relates to the magnitude of $ \tau _{k}$ over its task motion range. But $ \theta _{0}$ biases $ \tau _{k}$ direction sense, and so $ \theta _{0}$ is more critical for energy-minimizing designs. Advantageously, the SA-MRR allows online repositioning of $ \theta _{0}$ for greatest reconfigurability.

Defining optimal $ \theta _{0}$ also relates to segregation safety. To set (or reset) $ \theta _{0}$ initially at SA-MRR modules requires advancing the joint angle to engage the brake at $ \theta _{b}=\theta _{0}$ before a task commences. Practically, then, the $ \theta _{0}$ search space $ [\underline{\theta },\overline{\theta }]$ depends on whether the task safety constraints allow any joint to move outside the task limits $ [\theta _{\text{min}},\theta _{\text{max}}]$ at the outset. If so, a spatial safety analysis determines $ [\underline{\theta },\overline{\theta }]$ .

To find a fully specified, complementary spring characteristic, we formulate energy-minimizing gradient optimization problem

(17) \begin{equation} \begin{array}{rrl} \text{minimize} & f(k_{s},\theta _{0}) & = \displaystyle \int _{t_{0}}^{t_{f}}{P^{\prime}_{m}}(t,k_{s},\theta _{0})\text{d}t\\[10pt] \text{with respect to} & k_{s},\theta _{0} & \\[3pt] \text{subject to} & k_{s} & \geq 0\\[3pt] & \underline{\theta } & \leq \theta _{0}\leq \overline{\theta }\\[3pt] & t_{0} & \leq t\leq t_{f}\\ \text{and} & |\tau _{m_{\text{max}}}| & \gt \text{max}(|{\tau^{\prime}_{m}}(t,k_{s},\theta _{0})|)\\[3pt] \end{array} \end{equation}

which can be solved using Matlab’s nonlinear optimization function fmincon. Optimization enables flexibility to tailor the design search specifically for the task-specific torque and safety conditions of each joint module and simultaneously avoids infeasible designs which violate $ |\tau _{m_{\text{max}}}|$ .

This second MWM strategy and method (Fig. 5) potentially always decreases the task-specific motor energy consumption. The highest task-specific energy savings are generally achieved when both $ k_{s}$ and $ \theta _{0}$ are available for design (Fig. 6) compared to $ \theta _{0}$ only. By experience, many tasks prescribe an optimized $ \theta _{0}$ toward either the upper or lower bound of its search space, and so higher energy savings are usually possible when $ \theta _{0}$ may be set initially outside the task boundaries, $ [\theta _{\text{min}},\theta _{\text{max}}]$ . For joints confined for safety to $ [\theta _{\text{min}},\theta _{\text{max}}]$ , the spring is connected the first time $ \theta _{0}$ is reached during the task and stays connected thereafter. This has the advantage of no startup time, although energy of the first cycle typically exceeds the settled energy of each successive cycle. Clearly, the energy benefits of this MWM strategy are accentuated for tasks that repeat very often or endlessly. Experience and intuition agree that lower $ k_{s}$ shows better reconfigurability when gravity compensation is a large portion of the actuation energy. The energy characteristics of this safety-aware MWM energy efficiency method and of the energy-aware MWM safety method are demonstrated next in a simulated manipulation task.

Figure 5. Flowchart illustration of the proposed energy efficiency method.

Figure 6. Elbow joint torque–angle profile (solid blue) for 3-DoF SA-MRR task with spring-assisted torque (dash-dot red) that minimizes ${E^{\prime}_{m}}$ per cycle.