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Systematic variation of forcing parameters reveals complex dynamics which are sensitive to amplitude, phase and frequency.

# Introduction

• 本文分析简单运动模型的意义：为复杂设备的简单控制提供指导，揭露生物运动的原理。

Simple models have the ability to be fully analyzed and can thus provide guidance for simplifying control of more complex devices, and even reveal principles of biological locomotion.

• 研究背景：在越过复杂地形时，生物启发的跳跃机器人是轮式机器人的一个替代。目前，跳跃的最优策略往往是基于经验来调试，瞬间行为(如离地跳跃)的动态系统分析(dynamics of transient behaviors)是相对少的。

In robotics, biologically inspired legged jumping robots have been constructed as an alternative to wheeled robots to better traverse rough terrain.

The initial movement strategies for optimal jumping are typically chosen by empirical tuning for steady state hopping or squat jumps. Systematic studies of the dynamics of transient behaviors, critical to issues of lift-off, are relatively scarce.

# Experiment and model

• 硬件设计：线性马达+弹簧+推力棒+空气轴承(摩擦小)

The robot consisted of a linear motor actuator with a series spring rigidly attached to the bottom end of the actuator’s lightweight thrust rod. The actuator was mounted to an air bearing which allowed for 1D, and nearly frictionless, motion.

• 马达(actuator)相对轴承(bearing)的力量有限，所以通过倾斜来减少重力，但我不太理解这番话蕴含的物理意思，无法想象轴承和马达的位置关系是什么。

Due to power limitations in the actuator, the bearing was inclined at 15 degree relative to the horizontal, reducing gravitational acceleration to 0.276g.

• 马达相对于驱动棒(thrust rod)底部的位置关系表达式，如下所示。

• 设计巧妙处，利用电路(open circuit)做传感器，统计跳跃时间和高度，简单且精度高。

To detect lift-off, a continuity sensor attached to the bottom of the metal spring measured an open circuit when the spring left the ground.

A constant coefficient of restitution of 0.8 (measured from experiment) modeled the collision of the spring with the ground.

# Lift off and jump height

• 只研究马达运行一次的跳跃表现。

Since we were interested in rapid jumps from rest, actuator forcing was then restricted to only one cycle (N = 1). We systematically examined jumping height for N = 1.

## 两种跳跃模式

• 单次跳(single jump)

In the single jump mode, the robot compressed the spring and was propelled into the air.

• 突突跳(stutter jump)

In the stutter jump mode, the robot performed a small initial jump followed by a larger second jump.

## Single jump vs Stutter jump

• 起跳时间：The time to lift-off was smaller for single jumps than stutter jumps.

# Theory of transient mixing

• 看似简单的等式一，因为参数 $$\alpha$$ 的线性，更像是一个分段线性动态系统 (piecewise linear dynamical systems)，包含一系列复杂的行为。

• 基于等式一，理论分析为什么最佳跳跃不是发生在谐振频率。

We are particularly interested in why optimal jumps occur only off resonance.

## Single jump

Moving off resonance, the prefactor favors higher f over lower, so the optimum f lies somewhat above $$f_{0}$$ (resonance). This argument holds regardless of A.

## Single jump vs Stutter jump

• 最佳突突跳(stutter jump)的频率带宽(narrow frequency bandwidth)比较窄。 > This sensitivity to proper timing explains the narrow frequency bandwidth required to achieve optimal jump heights using the stutter jump mode.
• 最佳突突跳的频率强依赖幅值，单次跳则没有明显依赖。 > A further consequence is a strong dependence of optimal f with respect to A.
• 和单次跳跃相比，突突跳达到相同高度的消耗能量要少一个数量级。 > Thus the stutter jump is energetically advantageous since it has a lower optimal f than the single jump. In fact, the stutter jump uses nearly an order of magnitude less power to achieve comparable jump height to the single jump.

# Conclusion

## Works have done

• 发现突突跳，使用的能量少，但是跳跃高度高，性价比高。 > The system becomes hybrid for certain parameters as a stutter jump emerges. This mode achieves comparable jump height but uses less power.
• 研究成果和已有结论相符 > Biologically, our model is in accord with a previous model of bipedal jumping which predicted that counter-movement achieves greater jump height than the squat jump. A quick single jump that resembles a squat jump is beneficial when a fast escape is essential, while a slower stutter jump similar to a counter-movement can achieve comparable jump height.

## Works to do

• 研究其他影响因素，如系统固有性质和环境的影响 > how other factors, intrinsic and environmental, affect optimal performance
• A non-sinusoidal actuation could improve jump height, take off time, or efficiency.
• Animals jump off compliant surfaces (like tree branches) and from deformable substrates (like sand).
• 如何结合 Single jump 和 Stutter jump，取得更优的表现。