Movement

Basic Movement

移动算法的结构图

2D

角色在2维空间移动，大多数3维游戏也可以看做二维的移动。

Statics

在处理角色的位置、转向这些数据使用的公式与算法叫做静态，因为这些数据不含有任何角色移动的数据。数据结构可以定义为：

class Static:
    position: Vector
    orientation: float

Kinematic

如果一个角色正在向一个方向移动，突然改变他的速度与方向，这看起来有点突兀。为了让这个运动更加的丝滑，不让角色加速太快。我们就需要一些算法去考虑角色当前的速度，使用合理的加速度去改变速度。

我们就要记录这个角色的速度与转向的速度(角速度)，我们可以定义数据结构为：

class Kinematic:
    position: Vector
    orientation: float
    velocity: Vector
    rotation: float

Kinematic Movement

Seek

给定一个角色的静态数据以及目标的静态数据。来计算角色到目标的方向以及一个直线速度。大概实现如下：

class KinematicSeek:
    character: Static
    target: Static

    maxSpeed: float

    function getSteering() -> KinematicSteeringOutput:
        result = new KinematicSteeringOutput()

        # Get the direction to the target.
        result.velocity = target.position - character.position

        # The velocity is along this direction, at full speed.
        result.velocity.normalize()
        result.velocity *= maxSpeed

        # Face in the direction we want to move.
        character.orientation = newOrientation(
        character.orientation,
        result.velocity)

        result.rotation = 0
        return result

这个算法可以用于追击的一些情况。但是我们如果要让角色到一个点，然后停下来。我们可以添加一个以目标点为圆心的半径。当我们的角色到这个半径内时，就停止移动。当我们的设置半径不合理时，太小了，会使角色一直到不了目标点，还会看到角色的抖动。这里有几种解决方式：

• 固定时间到达目标点，需要设置一个最大速度阀值
• 扩大半径

class KinematicSeek:
    character: Static
    target: Static

    maxSpeed: float

    function getSteering() -> KinematicSteeringOutput:
        result = new KinematicSteeringOutput()

        # Get the direction to the target.
        result.velocity = target.position - character.position

         # Check if we’re within radius.
        if result.velocity.length() < radius:
            # Request no steering.
            return null
       
        # We need to move to our target, we’d like to
        # get there in timeToTarget seconds.
        result.velocity /= timeToTarget
       
        # If this is too fast, clip it to the max speed.
        if result.velocity.length() > maxSpeed:
            result.velocity.normalize()
            result.velocity *= maxSpeed

        # Face in the direction we want to move.
        character.orientation = newOrientation(
        character.orientation,
        result.velocity)

        result.rotation = 0
        return result

Wandering

Wandering（漫游）是一种让角色产生平滑随机游走效果的转向行为，其核心思想是：在角色前方一段固定距离处投射一个小圆（wander circle），目标点始终被约束在该圆的圆周上，每帧对目标点的角度施加一个小幅随机扰动（wander angle jitter），从而驱动角色连续、平滑地改变方向。

与简单地"每帧随机选一个新方向"相比，Wandering 的优势在于：随机量只作用于圆上的角度偏移，而非整体速度方向，因此方向变化是渐进的，视觉上不会产生抖动或突然转弯的不自然感。

class Wander:
    character: Kinematic

    # 投射到角色前方的距离
    wanderOffset: float
    # 投射圆的半径
    wanderRadius: float
    # 每帧最大角度扰动量（弧度）
    wanderRate: float
    # 当前目标点在圆上的角度（随帧累积）
    wanderOrientation: float

    maxAcceleration: float

    function getSteering() -> SteeringOutput:
        result = new SteeringOutput()

        # 在当前朝向上随机扰动 wander 角度
        wanderOrientation += randomBinomial() * wanderRate

        # 目标朝向 = 角色朝向 + 本帧 wander 角度
        targetOrientation = wanderOrientation + character.orientation

        # 投射圆的圆心（角色前方 wanderOffset 处）
        wanderCenter = character.position +
            wanderOffset * asVector(character.orientation)

        # 目标点 = 圆心 + wanderRadius * 目标朝向对应的单位向量
        target = wanderCenter + wanderRadius * asVector(targetOrientation)

        # 以最大加速度朝目标点施力
        result.linear = maxAcceleration * (target - character.position).normalized()
        result.angular = 0
        return result

其中 randomBinomial() 返回 $[-1,1]$ 范围内的随机值（通常用两个 $[0,1]$ 随机数相减近似正态分布），asVector(orientation) 将朝向角转换为单位向量。

Arrive

可以设置以目标点设置两个圈，一个大半径圈，当进入这个圈后，角色开始减速。小圈是判断角色是否已经到达。

class Arrive:
    character: Kinematic
    target: Kinematic
    maxAcceleration: float
    maxSpeed: float
    
    # The radius for arriving at the target.
    targetRadius: float
    
    # The radius for beginning to slow down.
    slowRadius: float
    
    # The time over which to achieve target speed.
    timeToTarget: float = 0.1
    
    function getSteering() -> SteeringOutput:
        result = new SteeringOutput()

        # Get the direction to the target.
        direction = target.position - character.position
        distance = direction.length()

        # Check if we are there, return no steering.
        if distance < targetRadius:
            return null
        
        # If we are outside the slowRadius, then move at max speed.
        if distance > slowRadius:
            targetSpeed = maxSpeed
        # Otherwise calculate a scaled speed.
        else:
            targetSpeed = maxSpeed * distance / slowRadius
        
        # The target velocity combines speed and direction
        targetVelocity = direction
        targetVelocity.normalize()
        targetVelocity *= targetSpeed
        
        # Acceleration tries to get to the target velocity.
        result.linear = targetVelocity - character.velocity
        result.linear /= timeToTarget
        
        # Check if the acceleration is too fast.
        if result.linear.length() > maxAcceleration:
            result.linear.normalize()
            result.linear *= maxAcceleration
        
        result.angular = 0

        return result

在很多实现中并没有使用到大圈这种方式，因为在减速是会有震荡的可能性

ALIGN

对齐使角色的转向与目标的转向相匹配。

class Align:
    character: Kinematic
    target: Kinematic

    maxAngularAcceleration: float
    maxRotation: float

    # The radius for arriving at the target.
    targetRadius: float

    # The radius for beginning to slow down.
    slowRadius: float

    # The time over which to achieve target speed.
    timeToTarget: float = 0.1

    function getSteering() -> SteeringOutput:
    result = new SteeringOutput()

    # Get the naive direction to the target.
    rotation = target.orientation - character.orientation

    # Map the result to the (-pi, pi) interval.
    rotation = mapToRange(rotation)
    rotationSize = abs(rotation)

    # Check if we are there, return no steering.
    if rotationSize < targetRadius:
        return null

    # If we are outside the slowRadius, then use maximum  tation.
    if rotationSize > slowRadius:
        targetRotation = maxRotation
     # Otherwise calculate a scaled rotation.
    else:
        targetRotation = maxRotation * rotationSize / slowRadius

    # The final target rotation combines speed (already inthe
    # variable) and direction.
    targetRotation *= rotation / rotationSize

    # Acceleration tries to get to the target rotation.
    result.angular = targetRotation - character.rotation
    result.angular /= timeToTarget

    # Check if the acceleration is too great.
    angularAcceleration = abs(result.angular)
    if angularAcceleration > maxAngularAcceleration:
        result.angular /= angularAcceleration
        result.angular *= maxAngularAcceleration

    result.linear = 0
        return result

VELOCITY MATCHING

把角色的速度与目标的数据设置为一样

class VelocityMatch:
    character: Kinematic
    target: Kinematic

    maxAcceleration: float

    # The time over which to achieve target speed.
    timeToTarget = 0.1

    function getSteering() -> SteeringOutput:
        result = new SteeringOutput()

        # Acceleration tries to get to the target velocity.
        result.linear = target.velocity - character.velocity
        result.linear /= timeToTarget

        # Check if the acceleration is too fast.
        if result.linear.length() > maxAcceleration:
            result.linear.normalize()
            result.linear *= maxAcceleration

        result.angular = 0
        return result

FACE

这个行为使角色看向目标,他委托对齐行为但是先计算目标的方向。

class Face extends Align:
    # Overrides the Align.target member.
    target: Kinematic

    # ... Other data is derived from the superclass ...

    # Implemented as it was in Pursue.
    function getSteering() -> SteeringOutput:
    # 1. Calculate the target to delegate to align
    Work out the direction to target.
    direction = target.position - character.position

    # Check for a zero direction, and make no change if so.
    if direction.length() == 0:
        return target

    # 2. Delegate to align.
    Align.target = explicitTarget
    Align.target.orientation = atan2(-direction.x, direction.z)
    return Align.getSteering()

LOOKING WHERE YOU`RE GOING

 class LookWhereYoureGoing extends Align:
    # No need for an overridden target member, we have
    # no explicit target to set.

    # ... Other data is derived from the superclass ...

    function getSteering() -> SteeringOutput:
        # 1. Calculate the target to delegate to align
        # Check for a zero direction, and make no change if so.
        velocity: Vector = character.velocity
        if velocity.length() == 0:
            return null

        # Otherwise set the target based on the velocity.
        target.orientation = atan2(-velocity.x, velocity.z)

        # 2. Delegate to align.
        return Align.getSteering()

WANDER