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// Copyright 2023 The Chromium Authors
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
#include "content/browser/navigation_transitions/physics_model.h"
#include <algorithm>
#include <numbers>
#include <vector>
#include "base/logging.h"
#include "base/notreached.h"
#include "base/numerics/ranges.h"
#include "base/time/time.h"
// TODO(liuwilliam): The velocity and positions should have the same direction.
//
// Notes:
// - Directions: for offsets/positions and the velocity of the cancel spring,
// the right edge direction is "+" and the left is "-"; for commit pending and
// invoke spring velocities, the right edge direction for is "-" and the left
// is "+".
// - The physics model internally operates in the normalized viewport space
// while takes/returns physical pixel values as input/output. The spacial
// variables are suffixed with `_viewport` or `_physical` to avoid confusion.
namespace content {
namespace {
// The tolerance value for which two floats are consider equal.
constexpr float kFloatTolerance = 0.001f;
// Used to replace NaN and Inf.
constexpr float kInvalidVelocity = 999.f;
// Springs.
//
// Determines when the spring is stabilized (the damped amplitude no longer
// changes significantly). Larger the value, the longer the spring takes to
// stabilize, but the spring amplitude is damped more gently.
constexpr int kSpringResponse = 708;
// How much the spring overshoots. Smaller the value, more bouncy the spring.
constexpr float kSpringDampingRatio = 0.81f;
// The size of the spring location history.
constexpr int kSpringHistorySize = 10;
// A spring is considered at rest if it has used at least
// `kSpringAtRestThreshold`*`kSpringHistorySize` amount of energy.
constexpr int kSpringAtRestThreshold = 10;
// Physics model.
//
// The size of the touch points store in `PhysicsModel`. Used to interpolate
// the finger's terminal velocity when the model switches from the finger drag
// curve driven to spring driven.
constexpr int kPhysicsModelHistorySize = 10;
bool IsValidVelocity(float velocity) {
return !base::IsApproximatelyEqual(velocity, kInvalidVelocity,
kFloatTolerance);
}
// Solves `positions`=`slope`*`timestamps`+ displacement(not calculated).
//
// TODO(crbug.com/40945408): The native least square might not give us
// the desired velocity.
void SolveLeastSquare(const std::vector<float>& timestamps,
const std::vector<float>& positions,
float* slope) {
CHECK_EQ(timestamps.size(), positions.size());
const size_t num_pts = timestamps.size();
if (num_pts <= 1) {
LOG(ERROR) << "Interpolating velocity with " << num_pts << " points";
if (slope) {
*slope = kInvalidVelocity;
}
return;
}
float sum_timestamps = 0;
float sum_positions = 0;
float sum_times_positions = 0;
float sum_timestamps_sq = 0;
for (size_t i = 0; i < num_pts; ++i) {
float t = timestamps[i];
float p = positions[i];
sum_timestamps += t;
sum_positions += p;
sum_times_positions += t * p;
sum_timestamps_sq += t * t;
}
if (slope) {
float denominator =
(sum_timestamps_sq - sum_timestamps * sum_timestamps / num_pts);
if (base::IsApproximatelyEqual(denominator, 0.f, kFloatTolerance)) {
*slope = kInvalidVelocity;
} else {
*slope =
(sum_times_positions - sum_timestamps * sum_positions / num_pts) /
denominator;
}
}
}
} // namespace
class Spring {
public:
struct Position {
// Calculated offset of the spring's position w.r.t. its equilibrium.
float equilibrium_offset_viewport;
// The amount of time delta since the spring is released (i.e., the start of
// the animation).
base::TimeDelta timestamp;
// If the spring is at rest then it won't bounce anymore. A spring is at
// rest if it has lost enough energe, or it is <= 1 pixel away from its
// equilibrium.
bool at_rest;
};
Spring(int frequency_response,
float damping_ratio,
float device_scaling_factor)
: frequency_response_(frequency_response),
damping_ratio_(damping_ratio),
device_scale_factor_(device_scaling_factor) {
float stiffness =
std::pow(2 * std::numbers::pi_v<float> / frequency_response_, 2) *
mass_;
undamped_natural_frequency_ = std::sqrt(stiffness / mass_);
damped_natural_frequency_ =
undamped_natural_frequency_ *
std::sqrt(std::abs(1 - std::pow(damping_ratio_, 2)));
// `damped_natural_frequency_` will be used as a denominator. It shouldn't
// be zero.
CHECK(!base::IsApproximatelyEqual(damped_natural_frequency_, 0.f,
kFloatTolerance));
CHECK(!base::IsApproximatelyEqual(device_scale_factor_, 0.f,
kFloatTolerance));
}
Spring(const Spring&) = delete;
Spring& operator=(const Spring&) = delete;
~Spring() = default;
Position GetPosition(float start_offset, base::TimeDelta time) {
// The general solution to a damped oscillator.
const float a = undamped_natural_frequency_ * damping_ratio_;
const float c =
(initial_velocity_ + a * start_offset) / damped_natural_frequency_;
const float ms = time.InMillisecondsF();
const float offset =
std::exp(-a * ms) *
(c * std::sin(damped_natural_frequency_ * ms) +
start_offset * std::cos(damped_natural_frequency_ * ms));
spring_position_history_.push_back({.equilibrium_offset_viewport = offset,
.timestamp = time,
.at_rest = false});
if (spring_position_history_.size() > kSpringHistorySize) {
spring_position_history_.pop_front();
float energy = 0;
for (const auto& p : spring_position_history_) {
// Energy is proportional to the square of the amplitude.
energy += p.equilibrium_offset_viewport * p.equilibrium_offset_viewport;
}
// If the spring has used `kSpringAtRestThreshold * kSpringHistorySize`
// amount energy in the last `kSpringHistorySize` locations, consider it
// is at rest.
spring_position_history_.back().at_rest |=
energy < kSpringAtRestThreshold * kSpringHistorySize;
}
// Less than 1pixel from its equilibrium.
spring_position_history_.back().at_rest |=
offset <= 1.f / device_scale_factor_;
return spring_position_history_.back();
}
float ComputeVelocity() {
std::vector<float> timestamps;
std::vector<float> positions;
timestamps.reserve(spring_position_history_.size());
positions.reserve(spring_position_history_.size());
for (const auto& p : spring_position_history_) {
timestamps.push_back(p.timestamp.InMillisecondsF());
positions.push_back(p.equilibrium_offset_viewport);
}
float velocity = 0;
SolveLeastSquare(timestamps, positions, &velocity);
return velocity;
}
float initial_velocity() const { return initial_velocity_; }
void set_initial_velocity(float velocity) { initial_velocity_ = velocity; }
private:
// Intrinsic properties of the spring.
const int frequency_response_;
const float damping_ratio_;
const float device_scale_factor_;
float undamped_natural_frequency_;
float damped_natural_frequency_;
const float mass_ = 1.f;
// The initial velocity might not be zero: to enture the smooth animation
// hand-off from spring A to spring B, we might set B's initial velocity to
// A's terminal velocity.
float initial_velocity_ = 0.f;
// The last few positions of the spring. Used to interpolate the velocity. It
// has a max size of `kSpringHistorySize`.
std::deque<Position> spring_position_history_;
};
PhysicsModel::PhysicsModel(int screen_width_physical, float device_scale_factor)
: viewport_width_(screen_width_physical / device_scale_factor),
device_scale_factor_(device_scale_factor) {
spring_cancel_ = std::make_unique<Spring>(
/*frequency_response=*/200,
/*damping_ratio=*/0.9,
/*device_scaling_factor=*/device_scale_factor_);
spring_commit_pending_ = std::make_unique<Spring>(
/*frequency_response=*/kSpringResponse,
/*damping_ratio=*/kSpringDampingRatio,
/*device_scaling_factor=*/device_scale_factor_);
spring_invoke_ = std::make_unique<Spring>(
/*frequency_response=*/200,
/*damping_ratio=*/0.95, /*device_scaling_factor=*/device_scale_factor_);
}
PhysicsModel::~PhysicsModel() = default;
PhysicsModel::Result PhysicsModel::OnAnimate(
base::TimeTicks request_animation_frame) {
// `commit_pending_acceleration_start_` needs to be recorded before we switch
// to the next driver.
RecordCommitPendingAccelerationStartIfNeeded(request_animation_frame);
AdvanceToNextAnimationDriver(request_animation_frame);
base::TimeDelta raf_since_start =
CalculateRequestAnimationFrameSinceStart(request_animation_frame);
// Ask the animation driver for the offset of the next frame.
Spring::Position spring_position;
switch (animation_driver_) {
case Driver::kSpringCommitPending: {
spring_position = spring_commit_pending_->GetPosition(
viewport_width_ * kTargetCommitPendingRatio -
animation_start_offset_viewport_,
raf_since_start);
// Prevent overshoot the right edge.
foreground_offset_viewport_ = std::min(
viewport_width_, viewport_width_ * kTargetCommitPendingRatio -
spring_position.equilibrium_offset_viewport);
// https://crbug.com/326850774: The commit-pending spring can also
// overshoot the left edge.
foreground_offset_viewport_ = std::max(0.f, foreground_offset_viewport_);
break;
}
case Driver::kSpringInvoke: {
spring_position = spring_invoke_->GetPosition(
viewport_width_ - animation_start_offset_viewport_, raf_since_start);
// Prevent overshoot the right edge.
foreground_offset_viewport_ = std::min(
viewport_width_,
viewport_width_ - spring_position.equilibrium_offset_viewport);
// https://crbug.com/326850774: The invoke spring can also overshoot the
// left edge.
foreground_offset_viewport_ = std::max(0.f, foreground_offset_viewport_);
break;
}
case Driver::kSpringCancel: {
spring_position = spring_cancel_->GetPosition(
animation_start_offset_viewport_, raf_since_start);
// Prevent overshoot the left edge.
foreground_offset_viewport_ =
std::max(spring_position.equilibrium_offset_viewport, 0.f);
// https://crbug.com/326850774: The cancel spring can also overshoot the
// right edge.
foreground_offset_viewport_ =
std::min(viewport_width_, foreground_offset_viewport_);
break;
}
case Driver::kDragCurve: {
NOTREACHED();
}
}
foreground_has_reached_target_commit_pending_ |=
foreground_offset_viewport_ >=
kTargetCommitPendingRatio * viewport_width_;
last_request_animation_frame_ = request_animation_frame;
return Result{
.foreground_offset_physical =
foreground_offset_viewport_ * device_scale_factor_,
.background_offset_physical =
ForegroundToBackGroundOffset(foreground_offset_viewport_) *
device_scale_factor_,
// Done only if we have finished playing the terminal animations.
.done = (spring_position.at_rest &&
(animation_driver_ == Driver::kSpringInvoke ||
animation_driver_ == Driver::kSpringCancel)),
};
}
// Note: we don't call `StartAnimating()` with the drag curve because
// `timestamp` for the drag curve is not from the wallclock. The non-wallclock
// time shouldn't be stored as `animation_start_time_`.
PhysicsModel::Result PhysicsModel::OnGestureProgressed(
float movement_physical,
base::TimeTicks timestamp) {
CHECK_EQ(animation_driver_, Driver::kDragCurve);
const float movement_viewport = movement_physical / device_scale_factor_;
foreground_offset_viewport_ =
std::max(0.f, FingerDragCurve(movement_viewport));
touch_points_history_.push_back(TouchEvent{
.position_viewport = foreground_offset_viewport_,
.timestamp = timestamp,
});
if (touch_points_history_.size() > kPhysicsModelHistorySize) {
touch_points_history_.pop_front();
}
return Result{
.foreground_offset_physical =
foreground_offset_viewport_ * device_scale_factor_,
.background_offset_physical =
ForegroundToBackGroundOffset(foreground_offset_viewport_) *
device_scale_factor_,
.done = false,
};
}
void PhysicsModel::SwitchSpringForReason(SwitchSpringReason reason) {
switch (reason) {
case kGestureCancelled:
case kGestureInvoked: {
// The navigation just started by the caller in the same atomic callstack
// if the user decides to start the navigation. The navigation hasn't
// committed or been cancelled yet.
CHECK_EQ(navigation_state_, NavigationState::kNotStarted);
if (reason == kGestureCancelled) {
navigation_state_ = NavigationState::kCancelled;
}
if (reason == kGestureInvoked) {
navigation_state_ = NavigationState::kStarted;
}
// Next `OnAnimate()` call will switch to `kSpringCancel` or
// `kSpringCommitPending`.
break;
}
case kBeforeUnloadDispatched: {
navigation_state_ = NavigationState::kBeforeUnloadDispatched;
// On next `OnAnimate()`, `animation_driver_` will switch to
// `kSpringCommitPending`.
break;
}
case kBeforeUnloadShown: {
CHECK_EQ(navigation_state_, NavigationState::kBeforeUnloadDispatched);
navigation_state_ = NavigationState::kBeforeUnloadShown;
// On next `OnAnimate()`, `animation_driver_` will switch to
// `kSpringCancel`.
break;
}
case kBeforeUnloadAckProceed: {
CHECK_EQ(navigation_state_, NavigationState::kBeforeUnloadShown);
navigation_state_ = NavigationState::kBeforeUnloadAckedProceed;
// On next `OnAnimate()`, `animation_driver_` will switch to
// `kSpringCommitPending`.
break;
}
case kCancelledBeforeStart: {
navigation_state_ = NavigationState::kCancelled;
}
}
}
void PhysicsModel::OnNavigationFinished(bool committed) {
switch (navigation_state_) {
// For a gesture navigation that doesn't have a BeforeUnload handler.
case NavigationState::kStarted:
// It's possible the navigation commits so fast that the commit-pending
// spring hasn't played a single frame.
case NavigationState::kBeforeUnloadDispatched:
// A navigation starts after running the BeforeUnload handler.
case NavigationState::kBeforeUnloadAckedProceed: {
break;
}
// A navigation needs to start first.
case NavigationState::kNotStarted:
// Not reachable because the browser is waiting for the ack from the
// renderer.
case NavigationState::kBeforeUnloadShown:
// A cancelled navigation should never commit.
case NavigationState::kCancelled:
// A navigation can only commit (finish) once.
case NavigationState::kCommitted: {
NOTREACHED();
}
}
navigation_state_ =
committed ? NavigationState::kCommitted : NavigationState::kCancelled;
}
bool PhysicsModel::ReachedCommitPending() const {
return animation_driver_ == Driver::kSpringCommitPending &&
foreground_has_reached_target_commit_pending_;
}
void PhysicsModel::StartAnimating(base::TimeTicks time) {
animation_start_time_ = time;
animation_start_offset_viewport_ = foreground_offset_viewport_;
}
float PhysicsModel::ForegroundToBackGroundOffset(float fg_offset_viewport) {
float bg_offset_viewport = 0.f;
if ((animation_driver_ == Driver::kSpringCommitPending ||
animation_driver_ == Driver::kSpringInvoke) &&
foreground_has_reached_target_commit_pending_) {
// Do not bounce the background page when the foreground page has reached
// the commit-pending point, once we have switched to the commit-pending
// spring or the invoke spring.
return bg_offset_viewport;
}
// Maps:
// fg_offset_viewport 0 -> 0.85W -> W
// To:
// bg_offset_viewport -0.25W -> 0 -> 0
const float fg_commit_position_viewport =
viewport_width_ * kTargetCommitPendingRatio;
// If the foreground has passed the commit position, the background should be
// at origin.
if (fg_offset_viewport > fg_commit_position_viewport) {
return bg_offset_viewport;
}
const float fg_progress_to_commit_position =
fg_offset_viewport / fg_commit_position_viewport;
bg_offset_viewport = (1 - fg_progress_to_commit_position) *
kScreenshotInitialPositionRatio * viewport_width_;
return bg_offset_viewport;
}
float PhysicsModel::FingerDragCurve(float movement_viewport) {
return foreground_offset_viewport_ +
kTargetCommitPendingRatio * movement_viewport;
}
float PhysicsModel::CalculateVelocity(base::TimeTicks time) {
float velocity = 0;
std::vector<float> timestamps;
std::vector<float> positions;
timestamps.reserve(touch_points_history_.size());
positions.reserve(touch_points_history_.size());
for (const auto& p : touch_points_history_) {
timestamps.push_back((time - p.timestamp).InMillisecondsF());
positions.push_back(p.position_viewport);
}
SolveLeastSquare(timestamps, positions, &velocity);
return velocity;
}
void PhysicsModel::RecordCommitPendingAccelerationStartIfNeeded(
base::TimeTicks request_animation_frame) {
if (animation_driver_ == Driver::kSpringCommitPending &&
navigation_state_ == NavigationState::kCommitted) {
float vel = spring_commit_pending_->ComputeVelocity();
if (IsValidVelocity(vel) && vel > kFloatTolerance) {
// If the navigation is committed and `spring_commit_pending_` is moving
// at the opposite direction of the invoke animation, record the first
// requested frame's timestamp. This timestamp will be used to speed up
// the opposite-moving animation of the commit-pending spring. Since the
// navigation is committed, we should display the invoke animation as soon
// as possible.
if (commit_pending_acceleration_start_.is_null()) {
commit_pending_acceleration_start_ = request_animation_frame;
}
} else {
// `spring_commit_pending_` moves in the same direction as the invoke
// animation. Reset `commit_pending_acceleration_start_`.
commit_pending_acceleration_start_ = base::TimeTicks();
}
}
}
void PhysicsModel::AdvanceToNextAnimationDriver(
base::TimeTicks request_animation_frame) {
switch (animation_driver_) {
case Driver::kDragCurve: {
// We can only reach here for once, and once only.
CHECK(last_request_animation_frame_.is_null());
StartAnimating(request_animation_frame);
float finger_vel = CalculateVelocity(request_animation_frame);
if (navigation_state_ == NavigationState::kCancelled ||
navigation_state_ == NavigationState::kBeforeUnloadShown) {
animation_driver_ = Driver::kSpringCancel;
// The sign of the velocities from the drag curve and the cancel spring
// have opposite semantics, so we need to multiply by -1.
spring_cancel_->set_initial_velocity(-finger_vel);
} else if (navigation_state_ == NavigationState::kCommitted) {
animation_driver_ = Driver::kSpringInvoke;
spring_invoke_->set_initial_velocity(finger_vel);
} else {
CHECK(navigation_state_ == NavigationState::kStarted ||
navigation_state_ == NavigationState::kBeforeUnloadDispatched ||
// This can happen when the renderer sends the BeforeUnload ack
// back to the browser so fast, that the cancel spring hasn't
// played a single frame thus we are still at
//`Driver::kDragCurve`. Typically this happens when the renderer
// doesn't have a sticky UserActivation.
navigation_state_ == NavigationState::kBeforeUnloadAckedProceed);
animation_driver_ = Driver::kSpringCommitPending;
spring_commit_pending_->set_initial_velocity(finger_vel);
}
break;
}
case Driver::kSpringCommitPending: {
// It is rare but possible that we haven't played a single frame with
// commit-pending spring, where `last_request_animation_frame_` is null.
auto start_animating_raf = !last_request_animation_frame_.is_null()
? last_request_animation_frame_
: request_animation_frame;
if (commit_pending_acceleration_start_.is_null() &&
navigation_state_ == NavigationState::kCommitted) {
// Only switch from commit-pending spring to the invoke spring when:
// - The commit-pending is moving in the same direction as the invoke
// animation, for which `commit_pending_acceleration_start_` is null.
// - The navigation is committed.
StartAnimating(start_animating_raf);
animation_driver_ = Driver::kSpringInvoke;
spring_invoke_->set_initial_velocity(
spring_commit_pending_->ComputeVelocity());
} else if (navigation_state_ == NavigationState::kCancelled ||
navigation_state_ == NavigationState::kBeforeUnloadShown) {
StartAnimating(start_animating_raf);
animation_driver_ = Driver::kSpringCancel;
// TODO(crbug.com/40945408): Ditto.
spring_cancel_->set_initial_velocity(1.f);
} else {
// Keep running the commit-pending animation if:
// - The commit-pending animation is being accelerated, for which
// `last_request_animation_frame_` is non-null.
// - The on-going navigation hasn't reached its final state
// (`OnDidFinishNavigation()` not yet called).
// - If the browser has asked the renderer to run the BeforeUnload
// handler but the renderer hasn't ack'ed the message.
const bool commit_pending_being_accelerated =
(!last_request_animation_frame_.is_null() &&
navigation_state_ == NavigationState::kCommitted);
const bool nav_started =
navigation_state_ == NavigationState::kStarted ||
navigation_state_ == NavigationState::kBeforeUnloadAckedProceed;
const bool nav_requested =
navigation_state_ == NavigationState::kBeforeUnloadDispatched;
CHECK(commit_pending_being_accelerated || nav_started || nav_requested);
}
break;
}
case Driver::kSpringCancel: {
if (navigation_state_ == NavigationState::kBeforeUnloadAckedProceed) {
// We only switch away from `kSpringCancel` when the renderer has acked
// the BeforeUnload message and navigation should proceed. When the
// BeforeUnload message is dispatched to the renderer, `kSpringCancel`
// drives the animation. When the renderer acks to proceed the
// navigation, we switch from `kSpringCancel` to `kSpringCommitPending`.
// It's visually incorrect to use `last_request_animation_frame_` here.
// Say the last frame is animated by `kSpringCancel` at time T and the
// user interacts with the BeforeUnload prompt at T+10s to begin the
// navigation. It's incorrect to tell `kSpringCommitPending` to animate
// the first frame as if it has been 10 seconds since the last frame.
StartAnimating(request_animation_frame);
animation_driver_ = Driver::kSpringCommitPending;
// Set the initial velocity to zero because the commit-pending (or
// invoke) spring will move the active page across the entire viewport.
// A high velocity would make the animation look like it's skipping
// frames.
spring_commit_pending_->set_initial_velocity(0.f);
} else if (navigation_state_ == NavigationState::kCommitted)
[[unlikely]] {
// Also rare but possible (e.g., in tests) for the navigation to commit
// so fast that the commit-pending spring hasn't played a single frame,
// after BeforeUnload is executed with "proceed". Directly switch to the
// invoke spring in this case.
StartAnimating(request_animation_frame);
animation_driver_ = Driver::kSpringInvoke;
spring_invoke_->set_initial_velocity(0.f);
}
break;
}
// Shouldn't switch from the terminal states.
case Driver::kSpringInvoke:
return;
}
if (!IsValidVelocity(spring_invoke_->initial_velocity())) {
spring_invoke_->set_initial_velocity(-2.0);
}
if (!IsValidVelocity(spring_commit_pending_->initial_velocity())) {
spring_commit_pending_->set_initial_velocity(0.f);
}
if (!IsValidVelocity(spring_cancel_->initial_velocity())) {
spring_cancel_->set_initial_velocity(-1.f);
}
}
base::TimeDelta PhysicsModel::CalculateRequestAnimationFrameSinceStart(
base::TimeTicks request_animation_frame) {
// Shouldn't be called for the drag curve animation.
CHECK_NE(animation_driver_, Driver::kDragCurve);
base::TimeDelta raf_since_start =
request_animation_frame - animation_start_time_;
// Accelerate the commit-pending animation if necessary.
if (!commit_pending_acceleration_start_.is_null()) {
CHECK_EQ(navigation_state_, NavigationState::kCommitted);
CHECK_EQ(animation_driver_, Driver::kSpringCommitPending);
// Add a delta to all the left-moving frames. This is to "speed up" the
// spring animation, so it can start to move to the right sooner, to display
// the invoke animation.
//
// Ex:
// - request animation frame timeline: [37, 39, 41, 43, 45 ...]
// - raf timeline with the delta: [37, 41, 45, 49, 53 ...]
//
// So the net effect is the animation is sped up twice.
raf_since_start +=
(request_animation_frame - commit_pending_acceleration_start_);
}
return raf_since_start;
}
} // namespace content
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