With regard to the small basin of attraction of the passive limit cycles, it is important to start from a proper initial condition for stable walking. The present study investigates the passive dynamic behaviors of two-dimensional bipedal walkers of a compass gait model with different foot shapes. In order to find proper initial conditions for stable and unstable period-one gait limit cycles, a method based on solving the nonlinear equations of motion is presented as a boundary value problem (BVP). An initial guess is required to solve the related BVP that is obtained by solving an initial value problem (IVP). For parametric analysis purposes, a continuation method is applied. Simulation results reveal two, period-one gait cycles and the effects of parameters variation for all models.

In order to prevent robot manipulators from reaching singularities, the “distance” from the current configuration to a singular configuration should be measured. This paper presents a novel metric based on geometric average normalized volume spanned by weighted screws to measure closeness to singularities for both serial and parallel manipulators. The weighted screws are proposed to reinterpret the physical meaning of twists and wrenches, so the problem of inconsistent dimensions in the dot product of screws has been eliminated. Compared with other existing methods, the proposed metric can obtain an identical result for similar manipulators with different sizes. Furthermore, the metric is independent of the selection of base screws, which is very suitable for the overconstrained or lower mobility parallel manipulator whose base screws are not uniquely definite. Besides, the geometrical meaning of the metric is related to the dimensionless volume of a high dimensional polyhedron, and hence the metric is insensitive to screw magnitude.

In Eq. (22), the x symbol was incorrectly represented by an @ symbol, therefore the equation should correctly read as

\begin{eqnarray*} \dot x &=& {f_v}(x) + {g_v}(x)u,\\ {y_v} &=& y_v^a(x) - y_v^d(x). \end{eqnarray*}

Non-central catadioptric vision is widely used in robotics and vision but suffers from the lack of an explicit closed-form forward projection model (FPM) that relates a 3D point with its 2D image. The search for the reflection point where the scene ray is projected is extremely slow and unpractical for real-time applications. Almost all methods thus rely on the assumption of a central projection model, even at the cost of an exact projection.

Two recent methods are able to solve this FPM, presenting a quasi-closed form FPM. However, in the special case of spherical mirrors, further enhancements can be made. We compare these two methods for the computation of the FPM and discuss both approaches in terms of practicality and performance. We also derive new expressions for the FPM on spherical mirrors (extremely useful to robotics and graphics) which speed up its computation.

This paper introduces a solution to the problem of steering an aerodynamical system, with non-holonomic constraints superimposed on dynamic equations of motion. The proposed approach is a dimensionality reduction of the Optimal Control Problem (OCP) with heavy path constraints to be solved by Rapidly-Exploring Random Tree (RRT) algorithm. In this research, we formulated and solved the OCP with Euler–Lagrange formula in order to find the optimal-time trajectory. The RRT constructs a non-collision path in static, high-dense obstacle environment (i.e. heavy path constraint). Based on a real-world aircraft model, our simulation results found the collision-free path and gave improvements of time and fuel consumption of the optimized Hamiltonian-based model over the original non-optimized model.

In this article, a two-dimensional analysis of biped robot sliding dynamics is performed. First, the dynamics of a biped robot based on feet-slip are derived using the coulomb friction model. The state transition can be formulated in the centroid acceleration space whose diagram is defined as a “triangle of sliding friction” (TSF). The TSF's characteristics are explained by focusing on comparison with the cone of friction which has a similar state decision diagram. Next, for the behavioral simulation of a concrete model, a 2-DOF biped robot is used to analyze the sliding features in terms of the asymmetry of the dynamics of each leg. Finally, the sliding characteristics are applied to the two tasks of running and somersaulting. The results show the robot takes short rapid repetitive steps for running based on frictional asymmetry and theoretically based on torque asymmetry can make one revolution using the large angular momentum acquired during sliding motion.

Large rotational angles about two axes for parallel mechanisms (PMs) with two rotational and three translational (2R3T) degrees of freedom (DOFs) or two rotational and two translational (2R2T) DOFs are demanded in some industries, such as parallel machine tools and multi-axis 3D printing. To address the problem, this paper focuses on the structural synthesis of new 2R3T and 2R2T PMs with high rotational capability. First, two new moving platforms are proposed based on the concepts of decoupled and configurable design. By means of the proposed platforms and Lie group theory, a series of 2R2T and 2R3T PMs are synthesized. Then the inverse kinematics and velocity relationship of one of the synthesized 2R3T PMs are presented. Finally, the rotational capability of the same 2R3T PM is analyzed. The result shows that by means of actuation redundancy, the studied 2R3T PM indeed possesses the high rotational capability about two axes, even though interferences and singularities are taken into consideration.

Legged locomotion systems have been effective in numerous robotic missions, and such locomotion is especially useful for providing better mobility over irregular landscapes. However, locomotion capabilities of robots are often constrained by a limited range of gaits and the associated energy efficiency. This paper presents the design of a novel reconfigurable Klann mechanism capable of producing a variety of useful gait cycles. Such an approach opens up new research avenues, opportunities and applications. The position analysis problem that arises when dealing with reconfigurable Klann mechanisms was solved here using a bilateration method, which is a distance-based formulation. By changing the linkage configurations, our aim was to generate a set of useful gaits for a legged robotic platform. In this study, five gait patterns of interest were identified, analysed and discussed that validate the feasibility of our approach and considerably extend the capabilities of the original design.

This paper presents a novel motion planning approach inspired by the Dynamic Programming (DP) applicable to multi degree of freedom robots (mobile or stationary) and autonomous vehicles. The proposed discrete–time algorithm enables a robot to reach its destination through an arbitrary obstacle field in the fewest number of time steps possible while minimizing a secondary objective function. Furthermore, the resulting optimal trajectory is guaranteed to be globally optimal while incorporating state constraints such as velocity, acceleration, and jerk limits. The optimal trajectories furnished by the algorithm may be further updated in real time to accommodate changes in the obstacle field and/or cost function. The algorithm is proven to terminate in a finite number of steps without its computational complexity increasing with the type or number of obstacles. The effectiveness of the global and replanning algorithms are demonstrated on a planar mobile robot with three degrees of freedom subject to velocity and acceleration limits. The computational complexity of the two algorithms are also compared to that of an A*–type search.

Variable stiffness mechanisms are able to mechanically reconfigure themselves in order to adjust their system stiffness. It is generally accepted that only antagonistic designs, featuring quadratic springs, can produce linear spring-like behaviour (i.e., a linear relationship between the displacement and its resultant force). However, these antagonistic designs typically are not as energy efficient as series-based designs. In this work, we propose a novel variable stiffness mechanism that can achieve both linear-spring behaviour whilst maintaining an energy efficient characteristic. This paper will present the working principle, mechanical design and characterization of the joints stiffness properties (verified via experimental procedure). The pros and cons of this novel design with reference to the other Variable Stiffness Actuator (VSA) designs will be discussed based on experimental results and in the context of general machining tasks.

Differential kinematics is a traditional approach to linearize the mapping between the workspace and joint space. However, a Jacobian matrix cannot be inverted directly in redundant systems or in configurations where kinematic singularities occur. This work presents a novel approach to the solution of differential kinematics through the use of dual quaternions. The main advantage of this approach is to reduce “drift” error in differential kinematics and to ignore the kinematic singularities. An analytical dual-quaternionic Jacobian is defined, which allows for the application of this approach in any robotic system.

The six-legged robot Octopus is designed for nuclear disaster relief missions. When the robot suffers from failures, its performance can be significantly affected. Thus, fault tolerance is essential for walking and operating in environments inaccessible to humans. The current fault-tolerant gaits for legged robots usually either initially lock the entire broken leg or just abandon the broken leg, but then fail to take full advantage of the normal actuators on the broken leg and add extra constraints. As the number of broken legs increases, the robot will no longer be able to walk using the existing fault-tolerant gaits. To solve this problem, screw theory is used for analyzing the remaining mobility after failure. Based on the analysis, a method of motion planning through fault-tolerant Jacobian matrices, which are linear, is presented. This method can enable the robot to accomplish desired movement using broken legs along with other certain concomitant motions as compensation. Finally, experiments and simulations of multiple faults demonstrate the real effects on the Octopus robot.

Swing-leg retraction in walking is the slowing or reversal of the forward rotation of the swing leg at the end of the swing phase prior to ground contact. For retraction, a hip torque is often applied to the swing leg at about the same time as stance-leg push-off. Due to mechanical coupling, the push-off force affects leg swing, and hip torque affects the stance-leg extension. This coupling makes the energetic costs of retraction and push-off depend on their relative timing. Here, we find the energy-optimal relative timing of these actions. We first use a simplified walking model with non-regenerative actuators, a work-based energetic-cost, and impulsive actuations. Depending on whether the late-swing hip torque is retracting or extending (pushing the leg forward), we find that the optimum is obtained by applying the impulsive hip torque either following or prior to the impulsive push-off force, respectively. These trends extend to other bipedal models and to aperiodic gaits, and are independent of step lengths and walking speeds. In one simulation, the cost of a walking step is increased by 17.6% if retraction torque comes before push-off. To consider non-impulsive actuation and the cost of force production, we add a force-squared (F2) term to the work cost. We show that this cost promotes simultaneous push-off force and retracting torque, but does not change the result that any extending torque should come prior to push-off. A high-fidelity optimization of the Cornell Ranger robot is consistent with the swing-retraction trends from the models above.

Our complete dynamical simulation-model realistically describes the real low-cost hexapod walker robot Szabad(ka)-II within prescribed tolerances under nominal load conditions. This validated model is novel, described in detail, for it includes in a single study: (a) digital controllers, (b) gearheads and DC motors, (c) 3D kinematics and dynamics of 18 Degree of Freedom (DOF) structure, (d) ground contact for even ground, (e) sensors and battery model. In our model validation: (a) kinematical-, dynamical- and digital controller variables were simultaneously compared, (b) differences of measured and simulated curves were quantified and qualified, (c) unknown model parameters were estimated by comparing real measurements with simulation results and applying adequate optimization procedures. The model validation helps identifying both model's and real robot's imperfections: (a) gearlash of the joints, (b) imperfection of approximate ground contact model, (c) lack of gearhead's internal non-linear friction in the model. Modeling and model validation resulted in more stable robot which performed better than its predecessors in terms of locomotion.